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TOWARDS THE FINITE SLOPE PART FOR GLn

CHRISTOPHE BREUIL AND FLORIAN HERZIG

Abstract. Let L be a finite extension of Qp and n ≥ 2. We associate to a crysta-belline n-dimensional representation of Gal(L/L) satisfying mild genericity assump-tions a finite length locally Qp-analytic representation of GLn(L). In the crystallinecase and in a global context, using the recent results on the locally analytic soclefrom [BHS17a] we prove that this representation indeed occurs in spaces of p-adicautomorphic forms. We then use this latter result in the ordinary case to show thatcertain “ordinary” p-adic Banach space representations constructed in our previouswork appear in spaces of p-adic automorphic forms. This gives strong new evidenceto our previous conjecture in the p-adic case.

Contents

1. Introduction 11.1. Notation 41.2. Acknowledgements 52. Preliminaries 53. Universal unitary completions 124. A partial adjunction 205. The finite slope space in the generic crystabelline case 245.1. Local setup and results 245.2. Global applications 275.3. Special cases and examples 316. Ordinary representations 336.1. Local setup and results 336.2. Global applications 417. Errata for [BH15] 43References 44

1. Introduction

Let G be a unitary group over a totally real number field F+ which becomes GLn overa totally imaginary quadratic extension F of F+ and such that G(F+⊗QR) is compact.Then the p-adic Banach spaces of continuous functions C0(G(F+)\G(A∞F+), E) for E a(varying) finite extension of Qp can be seen as a p-adic analogue of the complex Hilbertspace L2(G(F+)\G(AF+)). Assume that all places v|p of F+ split in F and choose v|v

Date: August 17, 2018.The second author was partially supported by an NSERC grant and a Simons Fellowship (Simons

Foundation grant #504284).1

2 CHRISTOPHE BREUIL AND FLORIAN HERZIG

in F for each v|p. Choose also a prime-to-p compact open subgroup Up =∏v-p Uv of

G(A∞,pF+ ), then the Up-invariant vectors S(Up, E) := {f : G(F+)\G(A∞F+)/Up → E}

form admissible continuous representations of G(F+ ⊗Q Qp) ∼=∏v|p GLn(Fv) over E

(withG(F+⊗QQp) acting by right translation of functions). Moreover to any absolutely

irreducible automorphic Galois representation r : Gal(F/F )→ GLn(E) of level Up one

can associate a non-zero invariant closed subspace S(Up, E)[mr] of S(Up, E).

The continuous representations S(Up, E)[mr], as well as their locally Qp-analytic

vectors S(Up, E)Qp-an[mr] have attracted some attention over the past years (see e.g.

[Eme06b], [BH15], [CEG+16], [BHS17a] and all the references therein). With thenotable exception of GLn(Fv) = GL2(Qp) for all v|p (see for instance [CDP14] and[CEG+]), and despite several partial results, these representations remain mysterious,e.g. one doesn’t even know if they have finite length. In this article, we focus on the

locally Qp-analytic representations S(Up, E)Qp-an[mr] when r is crystalline at all p-adicplaces and we use the recent results and techniques in [BHS17a] and [Brea] to deter-

mine an explicit subrepresentation of S(Up, E)Qp-an[mr], which is, to the knowledge ofthe authors, the largest known subrepresentation so far. Going beyond this subrepre-sentation will almost certainly require (seriously) new ideas.

Start with an arbitrary finite extension L of Qp, and a crystalline representation

ρ : Gal(L/L) → GLn(E) (here Hom(L,E) has cardinality [L : Qp]) with distinct σ-Hodge–Tate weights for all embeddings σ : L ↪→ E and such that the eigenvaluesϕ1, . . . , ϕn of ϕ[L0:Qp] on Dcris(ρ) satisfy ϕiϕ

−1j 6∈ {1, p±[L0:Qp]} for all i 6= j, where L0

is the maximal unramified extension contained in L. Then one can associate to ρ acertain list of distinct irreducible constituents (the “locally analytic socle”) Csoc(ρ) ={C(walg,F)} depending on two parameters: walg, which is a permutation of the σ-Hodge–Tate weights for each σ : L ↪→ E, and F which is a refinement, i.e. an orderingof the eigenvalues ϕi. These two parameters (walg,F) satisfy a certain relation thatinvolves the Hodge filtration on Dcris(ρ) ⊗L0 L (see §5.1). In fact C(walg,F) is thesocle of a certain locally Qp-analytic principal series PS(walg,F) of GLn(L), see (5.4),and C(1,F) is the usual locally algebraic representation associated to ρ by the classicallocal Langlands correspondence (it is in Csoc(ρ) and doesn’t actually depend on F).

Then one defines a finite length admissible locally Qp-analytic representation Π(ρ)fs

of GLn(L) over E (“fs” for “finite slope”) as follows (see Definition 5.7):

(i) For any (walg,F) such that C(walg,F) ∈ Csoc(ρ), we let M(walg,F) be thelargest subrepresentation of PS(walg,F) such that none of the irreducible con-stituents of M(walg,F)/C(walg,F) is in Csoc(ρ).

(ii) For any C ∈ Csoc(ρ) we define the following amalgam over the common socle C

M(ρ)C :=⊕C

{M(walg,F) : C(walg,F) ∼= C}

and denote by Π(ρ)fsC the unique quotient of M(ρ)C with socle isomorphic to C.

(iii) We finally set Π(ρ)fs :=⊕

C∈Csoc(ρ) Π(ρ)fsC .

TOWARDS THE FINITE SLOPE PART FOR GLn 3

The representation Π(ρ)fs has socle⊕

C∈Csoc(ρ)C and in general is not multiplicity

free. We give two explicit examples for GL3(Qp) in §5.3. Note that Π(ρ)fs does notallow one to recover the Hodge filtration on Dcris(ρ)⊗L0 L, though it depends on it.

Now, for v|p let rv := r|Gal(Fv/Fv) (where r is automorphic of level Up as above) which

we assume crystalline satisfying the above conditions for each v|p. One of us conjec-

tured in [Bre15, Conj. 6.1] that ⊗v|pC(walg

v ,Fv)(εn−1) appears as a subrepresentation

of S(Up, E)Qp-an[mr] if and only if C(walgv ,Fv) ∈ Csoc(rv) for each v|p (see Conjecture

5.10, here (εn−1) is the twist by the (n− 1)th power of the p-adic cyclotomic character

ε on each factor C(walgv ,Fv)). This conjecture was proven in [BHS17a, Thm. 1.3] under

the usual Taylor–Wiles assumptions when Up is sufficiently small and r is residuallyabsolutely irreducible (see Theorem 5.11).

Theorem 1.1 (see Corollary 5.16). Suppose that [Bre15, Conj. 6.1] holds and for eachC = ⊗v|pCv with Cv ∈ Csoc(rv) let

nC := dimE HomG(F+⊗QQp)

(⊗v|pCv(ε

n−1), S(Up, E)Qp-an[mr])∈ Z>0.

Then there exists an injective G(F+ ⊗Q Qp)-equivariant linear map

(1.2)⊕

C=⊗Cv

(⊗v|p

Π(rv)fsCv

(εn−1))⊕nC ↪→ S(Up, E)Qp-an[mr].

In particular there exists a G(F+ ⊗Q Qp)-equivariant injection of admissible locally

Qp-analytic representations ⊗v|p

Π(rv)fs(εn−1) ↪→ S(Up, E)Qp-an[mr].

In fact, Theorem 1.1 extends verbatim (and by the same proof) to the case when rv iscrystabelline (and not just crystalline) for each v|p (satisfying conditions analogous tothe ones above). But in the crystalline case at least it becomes unconditional thanks to[BHS17a, Thm. 1.3] (under the hypothesis of loc.cit.). Note that the embedding (1.2)should be quite far from being an isomorphism in general, for instance because one can

reasonably expect that the locally Qp-analytic representation S(Up, E)Qp-an[mr] doesdetermine all the Galois representations rv for v|p, and we know that this isn’t the casefor the left-hand side of (1.2).

The proof of Theorem 1.1 proceeds as follows: (i) one first deduces from a generaladjunction formula (Proposition 4.8 in the text) that any equivariant homomorphism

⊗v|pCv(ε

n−1) ↪→ S(Up, E)Qp-an[mr] extends uniquely to an equivariant homomorphism

⊗v|pM(rv)Cv(ε

n−1) → S(Up, E)Qp-an[mr], (ii) one proves (using [Bre15, Conj. 6.1]) that

any such homomorphism necessarily factors through the quotient ⊗v|p

Π(rv)fsCv

(εn−1),

and is injective since it is injective in restriction to the socle ⊗v|pCv(ε

n−1). The proof of

Proposition 4.8 itself relies on the same strategy that was already used in the proof of[Brea, Thm. 1.3] (based on an extension of functional analysis results of Emerton).

Theorem 1.1 and Proposition 4.8 (on which it crucially relies) “unify” various resultsand give evidence to several conjectures in the literature, which is the main reason


why we wrote this article. First Proposition 4.8 generalises an adjunction formula ofBergdall and Chojecki ([BC18, Thm. B]) in the case of a Borel subgroup and a locallyalgebraic character. When n = 2 the representation Π(ρ)fs is exactly the representationΠ(Dcris(ρ)) in [Bre16, §4(9)]. In this case Theorem 1.1 was already proven by Dingin the setting of the completed H1 of unitary Shimura curves (see [Din, Thm. 6.3.7])by a different argument. Theorem 1.1 also gives evidence to [Brea, Conj. 6.1.1], which

implies in particular that each constituent of the form C(walgv ,Fv)(ε

n−1) (assumingthere is only one place v|p in F+ for simplicity) which is not in Csoc(rv) and where

walgv is a simple reflection appears in the socle of S(Up, E)Qp-an[mr]/C(1,Fv)(ε

n−1) (seethe end of §5.3). Finally, Theorem 1.1 allows us to give strong evidence to a previousconjecture of the two authors in the ordinary case ([BH15, Conj. 4.2.2]). Considerthe crystalline representation ρ above and assume moreover that L = Qp and that ρ isupper triangular. In that case we have a canonical refinement Fρ and in [BH15, §3.3] we

associated to ρ a finite length continuous admissible representation Π(ρ)ord of GLn(Qp)over E of the form Π(ρ)ord =

⊕w Π(ρ)Cρ,w−1 , where w runs over those w = walg such

that C(w,w(Fρ)) ∈ Csoc(ρ) and where each Π(ρ)Cρ,w−1 is indecomposable and is a

successive extension of certain unitary continuous principal series of GLn(Qp) over E.

Theorem 1.3 (see Theorem 6.25). In the setting of Theorem 1.1, assume moreoverthat p is totally split in F+ and that each rv for v|p is upper triangular. For eachw = (wv)v|p such that C(wv, wv(Frv)) ∈ Csoc(rv) let

nw := dimE HomG(F+⊗QQp)

(⊗v|pC(wv, wv(Frv))(ε


Then there exists an injective G(F+ ⊗Q Qp)-equivariant linear map⊕w=(wv)v

(⊗v|p

Π(rv)Crv ,w−1v

(εn−1))⊕nw ↪→ S(Up, E)[mr].

In particular there exists a G(F+⊗QQp)-equivariant injection of admissible continuous

representations ⊗v|p

Π(rv)ord(εn−1) ↪→ S(Up, E)[mr].

As above, recall that Theorem 1.3 becomes unconditional under the assumptions of[BHS17a, Thm. 1.3]. One way to prove Theorem 1.3 goes as follows: (i) one first provesthat the locally analytic vectors (Π(ρ)ord)Qp-an of Π(ρ)ord is a (closed) subrepresentation

of Π(ρ)fs (Proposition 6.18), (ii) one proves that the universal unitary completion of(Π(ρ)ord)Qp-an gives back Π(ρ)ord (Proposition 6.20), (ii) one then combines these tworesults with Theorem 1.1 to deduce Theorem 1.3. (We actually give an alternativeproof, see §6.2.)

Along the way, we carefully prove several unsurprising but useful technical results(e.g. Lemma 2.10 or Lemma 3.4), some of which having already been tacitly used inprevious references (e.g. in [BHS17b], [BHS17a]). We also provide a complete proofto the crucial Proposition 4.1 in the text which was already stated (but without acomplete proof) in [Brea, Prop. 6.3.3].

1.1. Notation. We let E/Qp be a finite extension and O its ring of integers. The fieldE will be the coefficient field for all representations and locally convex vector spaces,unless otherwise stated. In particular all (completed) tensor products of locally convex


vector spaces will be over E. All locally analytic manifolds will be assumed to beparacompact.

If V is a locally convex vector space then V ′b is its continuous dual with the strongtopology ([Sch02, §9]). A unit ball in a Banach space is any open bounded lattice (orequivalently the unit ball of some norm defining the Banach topology). If V is anyBH-space (see [Eme17, Def. 1.1.1]), then V denotes the latent Banach space structureon V ([Eme17, §1.1]).

If Vi are locally convex vector spaces, we write V1 ⊗π V2 for the tensor productequipped with the projective topology ([Sch02, §17]). If the Vi are of compact type orFrechet, then this agrees with the inductive topology and we just write V1⊗V2 ([Eme17,Prop. 1.1.31]).

If L is a finite extension of Qp we will tacitly identify (characters of) L× with

(characters of) W abL by local class field theory, normalised so that uniformisers cor-

respond to geometric Frobenius elements, and where WL is the Weil group of L andW abL its maximal abelian quotient. We let | · |L be the normalised absolute value

x ∈ L× 7→ p−[L:Qp]val(x), where val(p) = 1. For instance, for L = Qp the cyclotomiccharacter ε is identified with the character x 7→ x|x|Qp of Q×p .

If I is a finite set, we denote by #I its cardinality.All other notation will be introduced in the course of the text.

1.2. Acknowledgements. The second author thanks the Universities of Paris-Sudand Paris 6, where some of this work was carried out, for pleasant working conditions.

2. Preliminaries

We establish some results in non-archimedean functional analysis that we couldn’tfind in the literature.

In this section, K will denote a subfield of E containing Qp (in particular, K is a finiteextension of Qp). For a locally K-analytic group G let Repla.cG denote the category oflocally K-analytic representations of G on locally convex vector spaces of compact typeand RepadG the full subcategory of admissible locally K-analytic representations (see[ST02b], [ST03], [Eme17]). Recall that a continuous linear map f : V → W betweenlocally convex vector spaces is strict if the continuous bijection V/ ker(f) → im(f) is

a topological isomorphism. If V is a locally convex vector space, we denote by V itsHausdorff completion ([Sch02, Prop. 7.5]).

Lemma 2.1.

(i) If 0 → V ′f−→ V

g−→ V ′′ is a strict short left exact sequence of locally convex

vector spaces, then the sequence 0→ V ′f−→ V

g−→ V ′′ is strict exact.(ii) If W is a locally convex vector space the functor W ⊗π (−) (resp. W ⊗

π(−))

is exact (resp. left exact), meaning that it sends strict short exact sequences(resp. strict short left exact sequences) of locally convex vector spaces to strictshort exact sequences (resp. strict short left exact sequences).


(iii) Suppose we are given locally convex vector spaces Vi (i ∈ I) and Wj (j ∈ J).Then we have a natural isomorphism(∏

i∈IVi)⊗π

(∏j∈J

Wj

) ∼= ∏(i,j)∈I×J

Vi ⊗πWj .

Proof. (i) We first consider the case where g is surjective. On the level of vector spaceswe have a left exact sequence with continuous maps because projective limits are leftexact. The map f is strict by the correspondence between open lattices in a locallyconvex vector space and in its completion. The map g is strict because the universal

property of completion shows that V / im(f)→ V ′′ is a completion map, and completionmaps are strict ([Sch02, Prop. 7.5]).

In the general case we factor g as composition V � im(g) ↪→ V ′′ of a strict surjectionand a strict injection. It remains to note that the completion of a strict injection is astrict injection (by the same argument as before), and that the composition of a strictmap and a strict injection is still strict.

(ii) Consider the sequence from (i) with g surjective. Then 1⊗π f is strict by [Sch02,Cor. 17.5] and 1⊗π g is strict by the definitions and by [Sch02, Cor. 17.5] again. Thestatement for ⊗

πnow follows from part (i).

(iii) It suffices to show that (∏i∈I Vi) ⊗

πW ∼=

∏i∈I(Vi ⊗

πW ) for any locally convex

vector space W . Since both sides are complete and Hausdorff, it suffices to show that

both spaces are canonically isomorphic after passing to HomC0

O (−,M) for any O-torsion

module M (with discrete topology), where HomC0

O means the continuous O-linear maps.We easily check

HomC0

O

((∏i∈I

Vi)⊗πW,M

) ∼= HomC0

O

((∏i∈I

Vi)⊗π W,M

).

Given any element f of this space, it has to vanish on (∏S Λi×

∏I−S Vi)×Λ for some

finite subset S ⊆ I and some open lattices Λi in Vi (i ∈ S) and Λ in W . By linearity,f is zero on (

∏I−S Vi) ⊗π W . By (ii) the function f factors to give an element of

HomC0

O ((∏S Vi)⊗π W,M). Hence

HomC0

O

((∏i∈I

Vi)⊗πW,M

) ∼= lim−→S

HomC0

O

((∏i∈S

Vi)⊗πW,M

) ∼= ⊕i∈I

HomC0

O (Vi⊗πW,M),

where the direct limit is over finite subsets S of I. Replacing for a moment Vi byVi ⊗

πW and W by E we also get

HomC0

O

(∏i∈I

(Vi ⊗

πW),M) ∼= ⊕

i∈IHomC0

O (Vi ⊗πW,M).

By combining all the above isomorphisms (and using again that M is complete) we aredone. �

We recall that a Hausdorff locally convex vector space is said to be hereditarilycomplete if all its Hausdorff quotients are complete. This is true for Frechet andcompact type spaces, cf. [Eme17, Def. 1.1.39].


Corollary 2.2. The completion functor is exact on any short exact sequence of locallyconvex vector spaces whose middle term has the property that its completion is heredi-tarily complete. The completed tensor product is even exact in the full subcategories ofFrechet and compact type spaces.

Proof. The first part follows from Lemma 2.1(i) and its proof. The second part thenfollows from Lemma 2.1(ii), as both subcategories are stable under completed tensorproducts [Eme17, Prop. 1.1.32]. (In the case of Frechet spaces, see also [Sch11, Cor.4.14].) �

Lemma 2.3. Suppose that C is a finite category and F a functor from C to RepadG.If V is a compact type space, then we have a topological isomorphism

(colimF ) ⊗V ∼= colim(F ⊗V ).

Proof. In the category RepadG any homomorphism is strict. The functor RepadG →Repla.cG sending W to W ⊗V commutes with finite direct sums and cokernels byCorollary 2.2, hence also with finite colimits. �

We will apply Lemma 2.3 in the case of colimits indexed by a partially ordered set.

Corollary 2.4. Suppose that V , W , U are Hausdorff locally convex vector spaces suchthat V and V ′b are bornological, W is hereditarily complete, and U is complete. Then

for any continuous linear map f : W → U we have ker(1 ⊗πf) ∼= V ⊗

π(ker f), where 1

denotes the identity of V .

Proof. By assumption, the map f factors as W �W/ ker f → U , where X = W/ ker fis complete, the first map is a strict surjection and the second map i is a continuousinjection. By [Eme17, Prop. 1.1.26], the map 1 ⊗

πi is injective. The claim then follows

from Lemma 2.1(ii) applied to 0→ ker f → W → W/ ker f → 0. (Note that the proofof [Eme17, Prop. 1.1.26] uses also that V ′b is bornological, in applying [Sch02, Prop.7.16].) �

If X is a locally K-analytic manifold and V a Hausdorff locally convex vector space,we denote by CK-an(X,V ) the locally convex vector space of locallyK-analytic functionsfrom X to V ([FdL99, Satz 2.1.10], [ST02b, §2]).

Lemma 2.5. Suppose that X1, X2 are locally K-analytic manifolds and that V1, V2

are locally convex vector spaces of compact type. Then the natural map

CK-an(X1, V1) ⊗πCK-an(X2, V2)

∼−→ CK-an(X1 ×X2, V1 ⊗V2)(2.6)

f1 ⊗ f2 7→((x1, x2) 7→ f1(x1) ⊗ f2(x2)

)is an isomorphism.

Proof. Write Xi =∐j∈Ji Xij as a disjoint union of compact open subsets. Then

CK-an(Xi, Vi) ∼=∏j∈Ji C

K-an(Xij , Vi) (see [ST02b, §2]) and similarly for CK-an(X1 ×X2, V1 ⊗V2). By Lemma 2.1(iii) we are thus reduced to the case where X1, X2 are com-pact. In this case the lemma follows from [ST05, A.1, A.2] and [Eme17, Prop. 2.1.28]by checking on the dense set of functions fi(xi) = ϕi(xi)vi with ϕi ∈ CK-an(Xi, E) andvi ∈ Vi. �


Let H,G be locally K-analytic groups such that H is a cocompact closed subgroupof G. Let V ∈ Repla.cH, then we have an exact functor (IndGH −)K-an : Repla.cH →Repla.cG. Explicitly (see [FdL99, §4.1])

(IndGH V )K-an = {f ∈ CK-an(G,V ) : f(hg) = hf(g) ∀h ∈ H, g ∈ G}

with left action of G by right translation of functions. Moreover, choosing a locallyanalytic splitting s : H\G → G of the projection G → H\G ([FdL99, Satz 4.1.1]), we

obtain an isomorphism of locally K-analytic manifolds H×H\G ∼−→ G, (h, x) 7→ hs(x)and hence an isomorphism of locally convex vector spaces (with [Eme17, Prop. 2.1.28])

(2.7) (IndGH V )K-an ∼= CK-an(H\G,V ) ∼= CK-an(H\G) ⊗V.

Then exactness follows from the last assertion in Corollary 2.2.

Lemma 2.8. Suppose that for i = 1, 2 we are given locally K-analytic groups Giwith closed and cocompact subgroups Hi, as well as locally analytic representation Vi ∈Repla.cHi. Then V1 ⊗V2 ∈ Repla.c(H1 ×H2) and we have a natural isomorphism

(2.9) (IndG1×G2H1×H2

V1 ⊗V2)K-an ∼= (IndG1H1V1)K-an ⊗(IndG2

H2V2)K-an

in Repla.c(G1 ×G2).

Proof. By [Eme17, Prop. 3.6.18] applied to the inflations of Vi in Repla.c(H1 ×H2) wededuce that V1 ⊗V2 ∈ Repla.c(H1×H2). Fix now sections Hi\Gi → Gi, which induce asection (H1×H2)\(G1×G2)→ G1×G2. From (2.6) applied with Xi := Hi\Gi and from

(2.7) we can identify the two sides of (2.9) and see that f1 ⊗ f2 for fi ∈ (IndGiHi Vi)K-an

is identified with (g1, g2) 7→ f1(g1) ⊗ f2(g2) in (IndG1×G2H1×H2

V1 ⊗V2)K-an. In particular,

the map (2.9) is G1 ×G2-equivariant. �

We will now prove a compatibility of the construction of Orlik–Strauch [OS15], asextended in [Bre16, §2] and [BHS17a, Rk. 5.1.2], with respect to product groups.

Lemma 2.10. Suppose that for i = 1, 2 we are given a locally Qp-analytic group Giwhich is of the form

∏nij=1Gij(Kij), where Kij/Qp is finite and Gij is a split connected

reductive group over Ki (we assume p > 3 as in [OS15] if at least one Gij has factorsof type different from A). Suppose for each i that Pi ⊆ Gi is a parabolic subgroup, thatMi ∈ O

pialg (where pi is the Lie algebra of the locally-Qp-analytic group Pi and we use

the notation of [OS15]), and that πPi is an (admissible) smooth representation of finitelength of the Levi quotient of Pi. Then (see [OS15] or [Bre16, §2] for the notation)

(2.11) FG1P1

(M1, πP1) ⊗FG2P2

(M2, πP2) ∼= FG1×G2P1×P2

(M1 ⊗M2, πP1 ⊗ πP2).

Proof. We take K = Qp for the purpose of this proof. We let gi be the Lie algebra of

Gi and U(gi), U(pi) the enveloping algebras. We note that M := M1 ⊗M2 ∈ Op1×p2alg

and that π := πP1 ⊗ πP2 is an (admissible) smooth representation of finite length ofthe Levi quotient of P1 × P2. Choose finite-dimensional subspaces Wi ⊆ Mi that arepi-stable and generate Mi as U(gi)-module. Then W := W1 ⊗W2 has the analogousproperties for the product group. For any ∂i ∈ U(gi) ⊗U(pi) Wi we get the following


commutative diagram from Lemma 2.5 and Lemma 2.8:

(2.12)

(IndG1

P1W ′1 ⊗ πP1)Qp-an ⊗(IndG2

P2W ′2 ⊗ πP2)Qp-an ∼ //

� _

��

(IndG1×G2

P1×P2W ′ ⊗ π)Qp-an� _

��

CQp-an(G1,W′1 ⊗ πP1

) ⊗πCQp-an(G2,W

′2 ⊗ πP2

)

∂1 ⊗ ∂2��

∼ // CQp-an(G1 ×G2,W′ ⊗ π)

∂1⊗∂2��

CQp-an(G1, πP1) ⊗πCQp-an(G2, πP2

)∼ // CQp-an(G1 ×G2, π),

where the vertical maps in the bottom square are as defined in [OS15, §4.4]. Letφi : U(gi) ⊗U(pi) Wi � Mi denote the natural surjection, and note that the naturalsurjection φ : U(g1 × g2)⊗U(p1×p2) W � M is identified with φ1 ⊗ φ2. By consideringthe diagram (2.12) we deduce that the left-hand side of (2.11) is identified (inside thetop left of (2.12)) with the simultaneous kernel of all ∂1 ⊗ ∂2 for ∂1 ⊗ ∂2 in kerφ1 ⊗W2

and W1 ⊗ kerφ2, equivalently with the simultaneous kernel of all ∂1 ⊗ 1 ∈ kerφ1 ⊗ 1and 1 ⊗ ∂2 ∈ 1 ⊗ kerφ2 (where 1 denotes alternatively the identity map of (IndG2

P2W ′2⊗

πP2)Qp-an and (IndG1P1W ′1⊗πP1)Qp-an). Since the U(gi) are noetherian, we easily deduce

the claim from Corollary 2.4. �

If G is a locally K-analytic group and σ a continuous representation of G on a Banachspace, we denote by σ(G,K)-an the subspace of σ of locally K-analytic vectors for theaction of G, which carries a natural locally convex topology finer than the subspacetopology ([Eme17, Def. 3.5.3]). If the group G is clear from the context, we will writeσK-an instead of σ(G,K)-an. If X,Y are topological spaces we denote by C0(X,Y ) thespace of continuous maps from X to Y .

Lemma 2.13. Suppose that H,G are locally K-analytic group such that H is a closedsubgroup of G. Assume that there exists a compact open subgroup G0 of G such thatG = HG0. If σ is a continuous representation of H on a Banach space, then

(IndGH σ(H,K)-an)K-an ∼−→((IndGH σ)C

0)(G,K)-an

,

where (IndGH σ)C0

is the Banach space {f ∈ C0(G, σ) : f(hg) = hf(g) ∀h ∈ H, g ∈ G}with the supremum norm over G0 and left action of G by right translation of functions.

Proof. Note that the assumption is satisfied if G is the group of K-points of a connectedreductive group over K and H is a parabolic subgroup.

Recall from [Eme17, §2.1, §3.5] that we have a continuous injection σK-an ↪→ σand hence a continuous injection CK-an(G, σK-an) ↪→ C0(G, σ). This induces a con-

tinuous injection (IndGH σK-an)K-an ↪→ (IndGH σ)C0

of closed subspaces, which is clearlyG-equivariant. By passing to locally analytic vectors we get a continuous injection

i : (IndGH σK-an)K-an ↪→ ((IndGH σ)C0)K-an by Prop. 2.1.30, Prop. 3.5.6 and Thm. 3.6.12

in [Eme17].We now show that i is surjective. Let H0 := H∩G0. Note that restriction maps iden-

tify (IndGH σ)C0

with (IndG0H0σ)C

0and (IndGH σK-an)K-an with (IndG0

H0σK-an)K-an (equiv-

ariantly for the action of G0). Therefore we may assume, without loss of generality,that G is compact.


Suppose that f ∈ ((IndGH σ)C0)K-an. Choose an analytic open subgroup L ⊆ G which

is the K-points of an affinoid rigid analytic group variety L defined over K (we use the

notation of [Eme17, §2.1]) such that f is L-analytic, i.e. the orbit map L→ (IndGH σ)C0

of f is rigid analytic in the sense of [Eme17, Def. 2.1.9(ii)]. In particular, for eachg ∈ G, the map og : L → σ, l 7→ f(gl) is rigid analytic. Now choose an analytic opensubgroup M ⊆ H which is the K-points of an affinoid rigid analytic group variety Mover K such that (i) M ⊆ H ∩

⋂g∈G gLg

−1 and (ii) for any g ∈ G, the map M → L,

m 7→ g−1mg is rigid analytic (in particular this applies to the inclusion M ⊆ L byconsidering g = 1). This is possible since G is compact.

Define σM-rig := Crig(M, σ)∆1,2(M) as in [Eme17, §3.3]. Fix coset representatives

g1, . . . , gn of G/L and define maps fi : giL→ σM-rig by fi(gil)(m) := f(mgil) = mf(gil)

for l ∈ L, m ∈M . Then fi is rigid analytic, since Crig(giL,Crig(M, σ)) ∼= Crig(giL×M, σ)and the function (gil,m) 7→ f(mgil) = f(gim

gi l) = ogi(mgi l) is rigid analytic, where

mgi := g−1i mgi. It follows that the fi define a locally analytic function f : G→ σM-rig

whose composition with σM-rig ↪→ σ is f . Then the composition of f with σM-rig ↪→σK-an is the desired preimage of f .

We deduce that i : (IndGH σK-an)K-an → ((IndGH σ)C0)K-an is a continuous bijection. It

is a topological isomorphism of LB-spaces by Prop. 3.5.6 and Thm. 1.1.17 in [Eme17].�

Lemma 2.14. Suppose that for i = 1, 2 we are given locally K-analytic groups Gi,as well as admissible continuous representations σi of Gi on Banach spaces ([ST02a,§3]). Then σ1 ⊗σ2 is an admissible continuous representation of G1×G2 on a Banachspace, and we have a natural isomorphism

(2.15) (σ1 ⊗σ2)(G1 ×G2,K)-an∼= (σ1)(G1,K)-an ⊗(σ2)(G2,K)-an

in Repla.c(G1 ×G2).

Proof. We simplify notation “K-an = (G,K)-an” as in the proof of Lemma 2.13. Theadmissibility claim follows from [BH15, Lemma A.3] (by passing to compact opensubgroups that act unitarily). Now suppose that Hi ⊆ Gi are analytic open subgroupswhich are K-points of affinoid rigid analytic group varieties Hi over K, so H1 ×H2 isan analytic open subgroup of G1 ×G2 (= K-points of H1 ×K H2). Then, by definition([Eme17, §3.3])

(2.16) (σ1 ⊗σ2)H1 × H2-rig∼= (Crig(H1, σ1) ⊗Crig(H2, σ2))∆1,2(H1)×∆1,2(H2).

We note that ∆1,2(Hi) acts continuously on Crig(Hi, σi), because it acts continuously onC0(Hi, σi) [Eme17, Prop. 3.1.5, 3.2.10] and therefore on C0(Hi, σi)Hi-rig

∼= Crig(Hi, σi)(see the comment after Def. 3.3.1 and also Prop. 3.3.7 in [Eme17]). (Alternatively,note that ∆1,2(Hi) acts continuously on Crig(Hi, E) ⊗ σi and hence on its completionCrig(Hi, σi).) As the Hi are topologically finitely generated, we easily deduce usingCorollary 2.2 and Corollary 2.4 (note that the strong dual of a Banach space is still aBanach space, hence is bornological)

(2.17) (Crig(H1, σ1) ⊗Crig(H2, σ2))∆1,2(H1)×∆1,2(H2)

∼= Crig(H1, σ1)∆1,2(H1) ⊗Crig(H2, σ2)∆1,2(H2) ∼= (σ1)H1-rig ⊗(σ2)H2-rig.


Finally we choose cofinal descending sequences H(j)i (j ≥ 1) of analytic open subgroups

of Gi (= K-points of affinoid group varieties H(j)i over K) as in the proof of [Eme17,

Prop. 6.1.3], so that the transition maps in (σi)K-an∼= lim−→j≥1

(σi)H(j)i -rig

are injective

and compact. By using the isomorphisms (2.16) and (2.17) together with [Eme17,Prop. 1.1.32] we deduce that (2.15) holds. �

We refer to [ST03, §3] and [Eme17, §1.2] for (weak) Frechet–Stein algebras andcoadmissible modules.

Lemma 2.18. Suppose that A, B are Frechet–Stein (E-)algebras. Then A ⊗B is aweak Frechet–Stein algebra, and if M is a coadmissible A-module and N is a coadmis-sible B-module, then M ⊗N is a coadmissible A ⊗B-module.

Proof. We start by choosing Frechet–Stein structures A ∼= lim←−An, B ∼= lim←−Bn in

the sense of [Eme17, Def. 1.2.10]. We note that A ⊗B ∼= lim←−(An ⊗Bn) by [Eme17,

Prop. 1.1.29] (and cofinality) and claim that this expression gives a weak Frechet–Steinstructure on A ⊗B in the sense of [Eme17, Def. 1.2.6]. The first two conditions of thatdefinition are verified, as An ⊗Bn is a Banach algebra. To check the third condition,it remains to show that the natural map A ⊗B → An ⊗Bn has dense image. Wewill show more generally that if Vi, V

′i (i = 1, 2) are locally convex vector spaces and

fi : Vi → V ′i are continuous maps with dense image, then f1 ⊗πf2 : V1 ⊗

πV2 → V ′1 ⊗

πV ′2

has dense image. It suffices to show that f1 ⊗π f2 : V1 ⊗π V2 → V ′1 ⊗π V ′2 has denseimage. By factoring f1 ⊗π f2 = (f1 ⊗π 1) ◦ (1 ⊗π f2) we moreover reduce to the casewhere V2 = V ′2 and f2 = id (as the composition of continuous maps with dense imageshas dense image). It is enough to show that for any open lattices Λ′1 of V ′1 and Λ2 of V2

we have im(f1)⊗V2 + Λ′1⊗Λ2 = V ′1 ⊗V2. Given v′1⊗ v2 ∈ V ′1 ⊗V2, choose a ∈ E× suchthat av2 ∈ Λ2 and write a−1v′1 = f1(v1) + x1 with v1 ∈ V1, x1 ∈ Λ′1 (which is possibleby assumption). Then v′1 ⊗ v2 = f1(v1) ⊗ av2 + x1 ⊗ av2 ∈ im(f1) ⊗ V2 + Λ′1 ⊗ Λ2, asrequired.

To check that M ⊗N is coadmissible, we write Mn := An ⊗AM(∼= An ⊗AM), Nn :=

Bn ⊗B N(∼= Bn ⊗BN) (which are Banach spaces by [ST03, Cor. 3.1]) and note that

M ⊗N ∼= lim←−(Mn ⊗Nn) by [Eme17, Prop. 1.1.29]. By assumption, Mn (resp. Nn) is a

finitely generated (locally convex) topological module over An (resp. Bn) in the senseof [Eme17, §1.2]. It follows from Corollary 2.2 that Mn ⊗Nn is a finitely generatedtopological An ⊗Bn-module. We have

(2.19) (An ⊗Bn) ⊗An+1 ⊗Bn+1

(Mn+1 ⊗Nn+1) ∼= (An ⊗An+1

Mn+1) ⊗(Bn ⊗Bn+1

Nn+1)

as topological An ⊗Bn-modules, because all spaces in question are Frechet (even Ba-nach), so using for instance Corollary 2.2 one easily checks that either side representsthe Hausdorff quotient of An ⊗Mn+1 ⊗Bn ⊗Nn+1 on which the two natural actionsof An+1, Bn+1 agree (acting on An, Bn on the right and Mn+1, Nn+1 on the left).Thus the module in (2.19) is isomorphic to Mn ⊗Nn. This completes the proof ofcoadmissibility. �


Lemma 2.20. Suppose that for i = 1, 2 we are given locally K-analytic groups Gi,as well as admissible locally K-analytic representations σi of Gi. Then the locallyK-analytic representation σ1 ⊗σ2 of G1 ×G2 is admissible.

Proof. For a locally K-analytic group G let DK-an(G) := CK-an(G,E)′b denote thelocally K-analytic distribution algebra of G. Let now G := G1 × G2. We may as-sume without loss of generality that both Gi are compact, and we need to show that(σ1 ⊗σ2)′b

∼= (σ1)′b ⊗(σ2)′b is a coadmissible DK-an(G)-module (the isomorphism followsfrom [Eme17, Prop. 1.1.32(ii)]). By the argument in Step 2 of the proof of [ST03,Thm. 5.1] we have a quotient map DQp-an(G)� DK-an(G) and it follows that a locallyK-analytic representation of G is admissible if and only if it is admissible as locallyQp-analytic representation (see the proof of [ST03, Prop. 3.7]). Thus we are reducedto the case where moreover K = Qp.

From Lemma 2.5 and [Eme17, Prop. 1.1.32(ii)] we have an isomorphism of Frechet–Stein algebras DQp-an(G) ∼= DQp-an(G1) ⊗DQp-an(G2), identifying a Dirac distributionδ(g1,g2) with δg1 ⊗ δg2 . The result then follows from Lemma 2.18. �

The following technical lemma will be needed later. We denote by C0G(X,Y ) the

space of continuous G-equivariant maps from X to Y .

Lemma 2.21. Suppose that Gi (i = 1, 2) are topological groups and that πi and Πi arelocally convex vector spaces equipped with a topological action of Gi such that the Πi

are of compact type. If C0G1

(π1,Π1) = 0, then C0G1×G2

(π1 ⊗ππ2,Π1 ⊗Π2) = 0.

Proof. Suppose that f ∈ C0G1×G2

(π1 ⊗ππ2,Π1 ⊗Π2). By assumption, for any y ∈ π2 and

any λ ∈ (Π2)′ we have ((1 ⊗λ) ◦ f)(π1 ⊗ y) = 0, so also ((1 ⊗λ) ◦ f)(π1 ⊗ππ2) = 0. It

thus suffices to show that if x ∈ Π1 ⊗Π2 and (1 ⊗λ)(x) = 0 for all λ ∈ (Π2)′, thenx = 0. By [Sch02, Cor. 18.8] we have Π1 ⊗Π2

∼= Lb((Π2)′b,Π1) (see loc.cit. for thenotation), and it is easily checked that this isomorphism is compatible with evaluationat any λ ∈ (Π2)′. The lemma follows. Note that the same argument shows in fact thatC0G1×G2

(π1 ⊗ιπ2,Π1 ⊗Π2) = 0, where π1 ⊗ι π2 is the tensor product equipped with the

inductive topology ([Sch02, §17]). �

3. Universal unitary completions

We compute the universal unitary completion of certain locally analytic parabolicinductions (Proposition 3.1).

We still denote by K a subfield of E containing Qp and by G the group of K-pointsof a connected reductive group over K. We refer to [Eme05, §1] for the definition ofuniversal unitary completions.

Proposition 3.1. Suppose that P is a parabolic subgroup of G with Levi subgroupM and that σ is a locally analytic representation of M of compact type satisfying thefollowing assumptions:

(i) σ admits a central character χσ;(ii) there exists a BH-subspace σ0 of σ such that σ =

∑m∈M mσ0.


Then σ has a universal unitary completion σ and the locally analytic representationπ := (IndGP σ)K-an satisfies the same hypotheses as σ (for G instead of M). If moreover

χσ is unitary, then the universal unitary completion π of π is given by π = (IndGP σ)C0,

together with the evident canonical map π → π.

Note that if χσ is non-unitary, then σ = 0, whereas π may be non-zero. For anexample, see the representation π in the proof of Proposition 6.6 below. Proposition3.1 has the following immediate corollary.

Corollary 3.2. Suppose that P is a parabolic subgroup of G with Levi subgroup Mand that σ is a finite-dimensional locally analytic representation of M such that σhas a unitary central character. Then (IndGP σ)K-an has universal unitary completion

(IndGP σ)C0.

Note that in the situation of Corollary 3.2, σ is the largest unitary quotient of theBanach representation σ and thus an isomorphism if and only if the M -action on σ isunitary. For example, if G is quasi-split, B = TU is a Borel subgroup, and χ a uni-

tary character of T , then the universal unitary completion of (IndGB χ)K-an is (IndGB χ)C0.

To prepare for the proof of Proposition 3.1, we first need some preliminary results.

Let P denote the parabolic subgroup opposite to P with common Levi subgroupM , and let N denote its unipotent radical. Choose z ∈ ZM , the centre of M . Choosean analytic open subgroup N0

∼−→ N0(K) of N (where as usual N0 is an affinoid rigidanalytic group over K) such that the function N0 → N0, n 7→ znz−1 is rigid analytic,i.e. lifts to N0 → N0. Recall [Eme17, §0.1] that a chart of a locally K-analytic manifoldX consists of an open subset X ′ ⊆ X, a K-affinoid closed ball X′, together with a locallyanalytic isomorphism X ′

∼−→ X′(K). For any chart φ : X∼−→ X(K) of N and any

m ∈ M , n ∈ N we denote for short by mnXm−1 the chart mnXm−1 ∼−→ Xφ−→ X(K),

where the first map sends mnxm−1 to x. Recall [Eme17, §2.1] that an analytic partitionof a locally K-analytic manifold X is a partition of X into a disjoint union of charts.The following lemma is clear.

Lemma 3.3. If⋂n≥0 z

nN0z−n = 1, then for any fixed m ≥ 0 the set {znνN0z

−n : ν ∈z−n−mN0z

n+m/N0}n≥0 is cofinal among all analytic partitions of z−mN0zm.

Lemma 3.4.

(i) Suppose for each 1 ≤ i ≤ r that Gi is the group of K-points of a connectedreductive group over K and that σi is a locally convex vector space equippedwith a continuous action of Gi such that σi has a universal unitary completionσi. Then the representation σ1 ⊗π · · · ⊗π σr of G1 × · · · × Gr has universalunitary completion σ1 ⊗ · · · ⊗ σr.

(ii) Suppose that H is the group of K-points of a connected reductive group over Kand that σ is a locally convex vector space equipped with a continuous actionof H such that σ has a universal unitary completion σ. Then the (usual)completion of σ has the same universal unitary completion.

Proof. (i) It is clear that G := G1 × · · · × Gr acts continuously on the locally convexvector space σ := σ1 ⊗π · · · ⊗π σr. We first show that for any G-invariant open lattice


Λ in σ there exist Gi-invariant open lattices Λi in σi such that Λ1⊗O · · ·⊗O Λr ⊆ Λ. Bydefinition of the projective tensor product topology, there exist open lattices Λ′i ⊆ σisuch that Λ′1 ⊗O · · · ⊗O Λ′r ⊆ Λ, so we can take Λi :=

∑gi∈Gi giΛ

′i. Hence, if Λi

for each i is a minimal (up to commensurability) Gi-stable open lattice in σi, thenΛ := Λ1⊗O · · ·⊗O Λr is a minimal (up to commensurability) G-stable open lattice in σ.Hence by [Eme05, Lemma 1.3] σ has universal unitary completion (σ1⊗· · ·⊗σr)Λ (with

the notation in [Sch02, §19]), which is isomorphic to σ1 ⊗ · · · ⊗ σr by [Sch02, Lemma19.10(ii)].

(ii) Let us write here σc for the usual completion of σ. Since H is locally compactit follows that H acts continuously on σc (use that a compact open subgroup of Hacts equicontinuously on σ [Eme17, §3.1]). If Λ is a minimal (up to commensurability)H-stable open lattice in σ, then its closure Λ is easily checked to be a minimal (up tocommensurability) H-stable open lattice in σc. Finally, observe that the natural mapσΛ → (σc)

Λ(again with the notation in [Sch02, §19]) is an isomorphism. �

Suppose that X is a locally K-analytic manifold and that σ is a Hausdorff locallyconvex vector space. We denote by C0

c(X,σ) the vector space of functions in C0(X,σ)that have compact support in X (recall the support is the closure of the non-vanishinglocus) and by CK-an

c (X,σ) the linear subspace of CK-an(X,σ) consisting of compactlysupported locally analytic functions, i.e. CK-an

c (X,σ) = CK-an(X,σ) ∩ C0c(X,σ). As in

[Eme07, §1] we give CK-anc (X,σ) the locally convex inductive limit topology according

to the isomorphism

(3.5) CK-anc (X,σ) ∼= lim−→

{Xi,Vi}i∈I

⊕i∈I

Crig(Xi, V i),

where {Xi∼−→ Xi(K)}i∈I runs through all analytic partitions of X and the Vi run

through all BH-subspaces of σ. The inclusion CK-anc (X,σ) ↪→ CK-an(X,σ) is continuous.

Moreover, the natural map

(3.6) lim−→V

CK-anc (X,V )→ CK-an

c (X,σ),

where V runs through all BH-subspaces of σ, is a continuous bijection. If X is compactwe have CK-an

c (X,σ) = CK-an(X,σ) and the map (3.6) is a topological isomorphism.Suppose that X is a reduced affinoid rigid analytic space over K. Let Crig(X, E)

denote the Banach algebra of E-valued rigid analytic functions on X, i.e. Crig(X, E) =A ⊗K E, where A is the affinoid algebra of X. Then Crig(X, E) is a reduced affinoidalgebra over E by [Con99, Lemma 3.3.1]. Since it is reduced, the usual supremumnorm defines the affinoid Banach topology. Recall also that if V is any Banach space(over E), then Crig(X, V ) denotes the Banach space Crig(X, E) ⊗V .

Definition 3.7.

(i) We denote by Crig(X,O) the open subring of power-bounded functions inCrig(X, E) (equivalently, the functions of supremum norm at most 1).

(ii) If V 0 is a unit ball of a Banach space V , we denote by Crig(X, V 0) the closureof Crig(X,O)⊗O V

0 inside Crig(X, V ).

Note that Crig(X, V 0) is a unit ball in Crig(X, V ). For example, if X is the n-dimensional closed unit ball, we can think of Crig(X, V ) as the Banach space of all


power series∑

i∈Nn Tivi with T i := T i11 · · ·T inn and vi ∈ V tending to zero in V as

i → ∞, and then Crig(X, V 0) consists of all such power series with vi ∈ V 0. If X is atopological space we remark that C0(X,V 0) is the space of functions in C0(X,V ) withimage in V 0. From Definition 3.7 we obtain the following lemma.

Lemma 3.8.

(i) The evaluation map Crig(X, V ) → C0(X(K), V ) restricts to Crig(X, V 0) →C0(X(K), V 0).

(ii) Suppose that V → W is any map of Banach spaces sending a unit ball V 0 ⊆V to a unit ball W 0 ⊆ W . Then the natural map Crig(X, V ) → Crig(X,W )restricts to Crig(X, V 0)→ Crig(X,W 0).

(iii) Suppose that X → Y is a map of reduced affinoid rigid analytic spaces overK. Then the natural map Crig(Y, V ) → Crig(X, V ) restricts to Crig(Y, V 0) →Crig(X, V 0).

Suppose now that X is a locally K-analytic manifold and V a Banach space. UsingLemma 3.8 we can make the following definition. We note from (3.5) that

CK-anc (X,V ) ∼= lim−→

{Xi}i∈I

⊕i∈I

Crig(Xi, V ),

where the inductive limit runs over all analytic partitions {Xi}i∈I of X and CK-anc (X,V )

is equipped with the locally convex inductive limit topology.

Definition 3.9. We let

CK-anc (X,V 0) := lim−→

{Xi}i∈I

⊕i∈I

Crig(Xi, V 0),

where the inductive limit runs over all analytic partitions {Xi}i∈I of X. If X is compactwe write CK-an(X,V 0) for CK-an

c (X,V 0).

If X ′∼−→ X′(K) is any chart of X, then the natural map Crig(X′, V ) → Can

c (X,V )is injective, since X′(K) is Zariski-dense inside X′. From the definitions it follows that

CK-anc (X,V ) =

∑X′ C

rig(X′, V ), where the sum runs over all charts X ′∼−→ X′(K).

Note also that CK-anc (X,V 0) is an open lattice in CK-an

c (X,V ) and that CK-anc (X,V 0) =∑

X′ Crig(X′, V 0) inside CK-an

c (X,V ). If X =∐i∈I Xi is a partition by open subsets,

then the natural map⊕

i∈I CK-anc (Xi, V )→ CK-an

c (X,V ) is a topological isomorphism,

identifying⊕

i∈I CK-anc (Xi, V

0) with CK-anc (X,V 0). (There are continuous maps in both

directions, using a cofinality argument.)

From Definition 3.9 and Lemma 3.8 we obtain the following lemma, whereC0c(X,V

0) := C0c(X,V ) ∩ C0(X,V 0).

Lemma 3.10.

(i) The natural injection CK-anc (X,V ) ↪→ C0

c(X,V ) restricts to CK-anc (X,V 0) ↪→

C0c(X,V

0).(ii) Suppose that V →W is any map of Banach spaces sending a unit ball V 0 ⊆ V

to a unit ball W 0 ⊆ W . Then the natural map CK-anc (X,V ) → CK-an

c (X,W )restricts to CK-an

c (X,V 0)→ CK-anc (X,W 0).


(iii) Suppose that X → Y is a proper map of locally K-analytic manifolds (i.e.,inverse images of compact sets are compact). Then the natural mapCK-anc (Y, V )→ CK-an

c (X,V ) restricts to CK-anc (Y, V 0)→ CK-an

c (X,V 0).

We next establish a compatibility of the lattices Crig(X, V 0) and CK-anc (X,V 0) with

respect to addition.

Lemma 3.11. Suppose that X is a reduced affinoid rigid analytic space over K.

(i) Suppose that V � W is a continuous surjection of Banach spaces, send-ing a unit ball V 0 ⊆ V onto a unit ball W 0 ⊆ W . Then the natural mapCrig(X, V 0)→ Crig(X,W 0) is surjective.

(ii) Suppose that σ is a Hausdorff locally convex vector space and that V1, V2 areBH-subspaces of σ with unit balls Λi ⊆ Vi. Then Λ1+Λ2 is a unit ball in V1 + V2

and we have Crig(X,Λ1)+Crig(X,Λ2) = Crig(X,Λ1+Λ2) inside Crig(X, V1 + V2).

Proof. (i) We equip Crig(X, E) (resp. V ) with the gauge norm of the open latticeCrig(X,O) (resp. V 0). Then Crig(X, V 0) is the unit ball in Crig(X, V ) = Crig(X, E) ⊗Vwith respect to the tensor product norm. Since the gauge norm of Crig(X, E) by defi-nition takes values in |E|E , it follows that Crig(X, E) is isometric to the Banach spacec0(I, E) of functions I → E that tend to zero, for some set I [Col10, I.1.5]. By [Col10,I.1.8] we have isometric isomorphisms Crig(X, V ) ∼= c0(I, E) ⊗V ∼= c0(I, V ), hence giv-ing a isomorphism Crig(X, V 0) ∼= c0(I, V 0) of unit balls. The claim now follows sincec0(I, V 0)→ c0(I,W 0) is clearly surjective.

(ii) By [Eme17, Prop. 1.1.5] note first that V1 + V2 is also a BH-subspace of σ. By[Eme17, Prop. 1.1.2(ii)] we have a continuous surjection of Banach spaces V1 ⊕ V2 �V1 + V2. It is an open map by the open mapping theorem and sends V 0 := Λ1 ⊕ Λ2

surjectively onto W 0 := Λ1 + Λ2, which is thus a unit ball in V1 + V2. We conclude by(i). �

Corollary 3.12. Suppose that X is a locally K-analytic manifold. Suppose that σ is aHausdorff locally convex vector space and that V1, V2 are BH-subspaces of σ with unitballs Λi ⊆ Vi. Then we have the following equalities inside CK-an

c (X,V1 + V2):

CK-anc (X,Λ1) + CK-an

c (X,Λ2) = CK-anc (X,Λ1 + Λ2),

CK-anc (X,V1) + CK-an

c (X,V2) = CK-anc (X,V1 + V2).

Proof. The first equality follows by summing the equality in Lemma 3.11(ii) over allcharts (or all analytic partitions) of X. The second follows by inverting p. �

Proposition 3.13. Suppose that X is a locally K-analytic manifold and V a Banachspace with unit ball V 0. Then we have inside C0

c(X,V ):

(3.14) CK-anc (X,V ) ∩ C0

c(X,V0) = CK-an

c (X,V 0).

Proof. It is clear that the right-hand side is contained in the left-hand side. To showthat the other inclusion holds, choose any analytic partition {Xi}i∈I of X. Then eachterm in (3.14) decomposes as a direct sum over i, and so we may assume without lossof generality that X ∼= OdK for some d ≥ 1.

We first make some preliminary remarks. Let $ denote a uniformiser of K, k theresidue field of K, q the cardinality of k, and $E a uniformiser of E. We equip theBanach space V with the gauge norm of V 0. Then any lift (vi)i∈I of an algebraic basis


of V 0/$EV0 is an orthonormal basis of V . Note that if X, Y are compact topological

spaces, then we have isomorphisms of Banach spaces (see e.g. [Sch02, §17])

C0(X,V ) ∼= C0(X,E) ⊗V and C0(X × Y,E) ∼= C0(X,E) ⊗C0(Y,E).

It is easy to verify that they are all isometries, where each function space carries thesupremum norm. On the other hand, for r ∈ |K×|K and a ∈ Kn let Br(a) denotethe K-affinoid closed ball of radius r and centre a. Then for a ∈ Kn, b ∈ Km

the canonical isomorphism Br(a) × Br(b) ∼= Br(a, b) induces an isometric isomor-phism Crig(Br(a, b), E) ∼= Crig(Br(a), E) ⊗Crig(Br(b), E). Define the Banach spaceLAh(OdK , E) :=

⊕Crig(B|$|hK (a), E), where the sum runs over representatives a of

OdK/$hOdK and each summand is equipped with the supremum norm. It follows from

what is before that LAh(OdK , E) ∼=⊗

1≤i≤d LAh(OK , E) is an isometric isomorphism.

Define more generally the Banach space LAh(OdK , V ) :=⊕

Crig(B|$|hK (a), V ), where

each summand is equipped with the tensor product norm. Hence its unit ball isLAh(OdK , V

0) :=⊕

Crig(B|$|hK (a), V 0), and we have a topological isomorphism

CK-an(OdK , V ) ∼= lim−→h

LAh(OdK , V ) ∼= lim−→h

(LAh(OdK , E) ⊗V )

which restricts to CK-an(OdK , V0) ∼= lim−→h

LAh(OdK , V0).

We now recall a particular Mahler basis of C0(OK , E) from [dS16, §1]. For m ≥ 1

let gqm(z) := $−(qm−1)/(q−1)∏r∈Rm(z − r), where Rm := {

∑m−1i=0 ai$

i : ai ∈ k} and

tilde denotes the Teichmuller lift. Any integer n ≥ 0 can be written n =∑m−1

i=0 biqi for

some m ≥ 1 and some bi ∈ {0, . . . , q−1}, and we define gn :=∏m−1i=0 gbi

qiand, for h ≥ 0,

gn,h := $∑s>hb

nqscgn. Then the {gn}n≥0 form an orthonormal basis of C0(OK , E), and

for any h ≥ 0, the {gn,h}n≥0 form an orthonormal basis of LAh(OK , E) (see Thm. 1.1and Prop. 4.2 in [dS16]).

Suppose now that f ∈ CK-an(OdK , V ) ∩ C0(OdK , V0). Then f ∈ LAh(OdK , V ) for some

h ≥ 0 and by what is above we can uniquely write

f(z1, . . . , zd) =∑n,i

λn,ign1(z1) · · · gnd(zd)vi,

where n = (n1, . . . , nd) ∈ Zd≥0, i ∈ I, λn,i ∈ O and µn,i,h := λn,i$−

∑j≤d;s>hb

njqsc → 0

in E. Then for all but finitely many pairs (n, i) ∈ Zd≥0 × I we have µn,i ∈ O. We can

therefore find an h′ ≥ h such that µn,i,h′ = λn,i$−

∑j≤d;s>h′b

njqsc ∈ O for all pairs (n, i).

It follows that f ∈ LAh′(OdK , V

0) ⊆ CK-an(OdK , V0), as required. �

Lemma 3.15. Suppose that σ is a locally analytic representation of G of compact typesuch that there exist a BH-subspace σ0 of σ and elements gi ∈ G for i ≥ 1 satisfyingσ =

∑∞i=1 giσ0. Then σ has a universal unitary completion. Explicitly, if Λ0 is any

unit ball in σ0, then Λ :=∑

g∈G gΛ0 is a minimal (up to commensurability) G-stableopen lattice in σ.

Proof. The proof only uses that σ is an LF-space [Eme17, Def. 1.1.16]. Since σ isan LF-space we can write σ as an increasing union of FH-subspaces σi (i ≥ 1) suchthat σ ∼= lim−→σi is a topological isomorphism. It is clear that Λ as in the statement


is a G-stable lattice in σ. We now verify that Λ is open. Let σ′i :=∑

j≤i gjσ0. Then

by assumption, σ is an increasing union of the FH-subspaces σ′i (using [Eme17, Prop.1.1.5]). By [Eme17, Prop. 1.1.10] the sequences σi and σ′i are mutually cofinal, and

we obtain that σ ∼= lim−→σ′i is a topological isomorphism. Therefore, to check that Λ is

open, it suffices to show that its preimage in σ′i is open for all i (see [Sch02, §5.E]).This is true since that preimage contains the lattice

∑j≤i gjΛ0, which is easily checked

to be open using the open mapping theorem.Suppose now that Ξ is any G-stable open lattice in σ. Then Ξ ∩ σ0 is open, hence

after scaling Ξ we may assume that Ξ ∩ σ0 ⊇ Λ0, so Ξ contains∑

g∈G gΛ0 = Λ. This

shows that Λ is a minimal (up to commensurability) G-stable open lattice in σ, andhence that σ has a universal unitary completion, namely σΛ. �

Proof of Proposition 3.1. Choose a cofinal sequence M1 ⊇ M2 ⊇ . . . of analytic opensubgroups of M and for each i let Mi be the affinoid rigid analytic K-group such thatMi = Mi(K).

Step 1: We show that we can enlarge our given σ0, if necessary, to ensure that thereexists an analytic open subgroup M ′0 of M such that σ0 is M ′0-stable and the inducedaction of M ′0 on σ0 is M′0-analytic (with obvious notation).

By assumption, σ ∼= lim−→i≥1τi, where τi is a Banach space and the transition maps

are injective (and compact). Let τi denote the image of τi in σ. By Propositions 3.2.15and 1.1.10 in [Eme17] (using the completeness of σ) and an easy induction we mayassume that each τi is M1-stable. Thus, as σ is a locally analytic representation of M ,by [Eme17, Thm. 3.6.12] we have topological isomorphisms

σ ∼= lim−→j

σMj-an∼= lim−→

j,i

(τi)Mj-an∼= lim−→

i

(τi)Mi-an.

Let σi := (τi)Mi-an and let σi denote its image in σ, a BH-subspace. By [Eme17, Prop.1.1.10] there exists an i ≥ 1 such that σ0 ⊆ σi. Then we can enlarge σ0 to σi and takeM ′0 := Mi.

Step 2: We show that σ admits a universal unitary completion and identify a minimal(up to commensurability) G-stable open lattice in σ.

By [Eme06a, Prop. 4.1.6] we may choose an analytic open subgroup H = H(K) of Gthat has a rigid analytic Iwahori decomposition with respect to P and P in the senseof [Eme06a, Def. 4.1.3]. This means in particular that there are closed rigid analyticsubgroups M0, N0, N0 of H satisfying M0(K) = H ∩M , N0(K) = H ∩ N , N0(K) =

H ∩N , such that the multiplication map induces an isomorphism N0×M0×N0∼−→ H

of rigid spaces. We let M0 := H∩M , N0 := H∩N , N0 := H∩N . Let AM ⊆M be thesplit part of the connected centre of M and choose z ∈ AM that strictly contracts eachroot subspace of the Lie algebra of N . By shrinking H, we may assume that M0 ⊆M ′0(as analytic open subgroups) and that the conjugation map N0 → N0, n 7→ znz−1 isrigid analytic.

Note that σ0 is an M0-analytic representation of M0, as M0 ⊆ M ′0. In particular,M0 acts continuously on σ0, so we can find an M0-stable unit ball Λ0 ⊆ σ0 [Eme17,Lemma 6.5.4]. Since M/M0 is countable, Lemma 3.15 implies that Λ :=

∑m∈M mΛ0

is a minimal (up to commensurability) M -stable open lattice in σ, and that σ ∼= σΛexists.


Pick coset representatives mi (i ≥ 1) for M/M0. In particular, Λ =∑miΛ0. We

define the increasing sequence of BH-subspaces σ′i :=∑

j≤imjσ0, so that σ ∼= lim−→σ′i(see the proof of Lemma 3.15).

Step 3: We set up some notation. Let π := (IndGP σ)K-an. Recall that CK-anc (N, σ)

is naturally a closed P -invariant subspace of π, with image consisting of all functionswhose support is contained in PN [Eme, Lemma 2.3.6]. We normalise the closedembedding CK-an

c (N, σ) ↪→ π so that the inverse map (on its image) is f ∈ π 7−→ (n ∈N 7→ f(n−1)). The P -action on CK-an

c (N, σ) is then given explicitly by (mnf)(n′) =

mf(n−1m−1n′m). In particular, for any chart X∼−→ X(K) of N and any BH-subspace

V of σ, Crig(X, V ) embeds as a BH-subspace of CK-anc (N, σ) and hence of π. For

m ∈ M , n ∈ N we see that mnCrig(X, V ) = Crig(mnXm−1,mV ) and in the limit thatmnCK-an

c (X,V ) = CK-anc (mnXm−1,mV ) (as subspaces of CK-an

c (N, σ)). Similarly, ifV 0 is a unit ball in V , then mnCrig(X, V 0) = Crig(mnXm−1,mV 0), where mV 0 denotesthe image of V 0 inside mV and mnCK-an

c (X,V 0) = CK-anc (mnXm−1,mV 0).

Step 4: We show that π satisfies hypotheses (i) and (ii) (for G instead of M). It isobvious that π has a central character. To verify hypothesis (ii), we take as BH-subspaceCrig(N0, σ0). By compactness of P\G we can find a finite partition P\G =

⊔sj=1Cj into

compact open subsets and elements gj ∈ G such that Cj ⊆ P\PNg−1j . It is enough to

show that

(3.16) π =∑

1≤j≤s,i≥1,n≥0,ν∈N/N0

gjmiznνCrig(N0, σ0)

with N0, σ0 and (mi)i≥1 as in Step 2. Inside CK-anc (N, σ) we have by the equalities in

Step 3 ∑n≥0,ν∈N/N0

znνCrig(N0, σ0) =∑n,ν

Crig(znνN0z−n, σ0) = CK-an

c (N,σ0),

as σ has a central character and using Lemma 3.3. From Corollary 3.12 and thebijectivity of (3.6) we then deduce∑

i≥1

miCK-anc (N,σ0) =

∑i≥1

CK-anc (N,miσ0) =

⋃`≥1

CK-anc (N,σ′`) = CK-an

c (N, σ)

with σ′` as in Step 2. (For the last equality note once again that the σ′` are cofinal among

all BH-subspaces of σ.) By partitioning the support of functions in π = (IndGP σ)K-an

and translating, it follows that π =∑s

j=1 gjCK-anc (N, σ). We obtain (3.16) by combin-

ing these equalities.

Step 5: We now assume that the central character χσ of σ is unitary. Define Λ :=∑g∈G gC

rig(N0,Λ0) and Λ′ := π ∩ (IndGP Λ)C0, where Λ =

∑m∈M mΛ0 (see Step 2) and

the intersection is taken inside (IndGP σ)C0. We will show that Λ = Λ′, and that it is a

minimal (up to commensurability) G-stable open lattice in π.According to equation (3.16) in Step 4 we may apply Lemma 3.15 to the BH-subspace

Crig(N0, σ0) of π and see that Λ is a minimal (up to commensurability) G-stable open

lattice in π. Since Crig(N0,Λ0) ⊆ C0(N0,Λ0) ⊆ (IndGP Λ)C0

(noting that MΛ0 ⊆ Λ), it

follows that Λ ⊆ Λ′, as Λ′ is G-stable by definition.


For the reverse inclusion, suppose that f ∈ Λ′. By partitioning the support off and by translating (as in Step 4) we may assume, without loss of generality, thatsupp(f) ⊆ PN0, so f ∈ CK-an(N0, σ) ∩ C0(N0,Λ). By Propositions 2.1.6 and 1.1.10

in [Eme17] and recalling that σ ∼= lim−→iσ′i, there exists i such that f ∈ C0(N0, σ′i). It

follows that im(f) ⊆ σ′i is compact. Recall that Λ′k :=∑

j≤kmjΛ0 is an open lattice

in σ′k (see the proof of Lemma 3.15), and note that Λ =⋃k Λ′k. Then for the value

of i above, Λ ∩ σ′i is an increasing union of the open lattices Λ′k ∩ σ′i for k ≥ i. By

compactness, im(f) ⊆ Λ′k ∩ σ′i ⊆ Λ′k for some k ≥ i. By increasing i, we conclude that

f ∈ CK-an(N0, σ′i) ∩ C0(N0,Λ′i).

By Proposition 3.13, we have f ∈ CK-an(N0,Λ′i) ⊆ CK-an

c (N,Λ′i). By Corollary 3.12,Lemma 3.3 and since σ has a unitary central character we have (as in Step 4)

f ∈∑j≤i

CK-anc (N,mjΛ0) =

∑j≤i

mjCK-anc (N,Λ0) =

∑ν∈N,j≥1,n≥0

mjCrig(znνN0z

−n,Λ0)

=∑

ν∈N,j≥1,n≥0

mjznνCrig(N0,Λ0),

and this is contained in Λ, as required.

Step 6: We show that π is canonically isomorphic to (IndGP σ)C0.

Consider the natural map θ : π = (IndGP σ)K-an → (IndGP σΛ)C0

(with Λ as in Step 2and Step 5). By choosing a locally analytic section of the map G → P\G, the map θis identified with CK-an(P\G, σ) → C0(P\G, σΛ). Since the locally constant, σ-valued

functions are dense in C0(P\G, σΛ), we deduce that θ has dense image. Moreover, if Λ

denotes the closure of the image of Λ in σΛ (whose preimage in σ is Λ), then Λ′ is the

preimage under θ of the unit ball (IndGP Λ)C0. It follows that θ induces a topological

isomorphism πΛ′

∼−→ (IndGP σΛ)C0. By Step 2 and Step 5 we know that σ ∼= σΛ and

π ∼= πΛ′

. This finishes the proof of Proposition 3.1. �

4. A partial adjunction

We prove an adjunction result (Proposition 4.8) which will be crucially used in theproof of the main result.

We first need to discuss some technical preliminaries. We let G :=∏di=1 GLn(Li)

with Li/Qp finite for all i. We let B (resp., N , resp., T ) denote the subgroup of upper-

triangular (resp., upper-triangular unipotent, resp., diagonal) matrices in G. Let b, tdenote the Qp-Lie algebras of B, T . Let N0 denote any compact open subgroup of N

and let T+ := {t ∈ T : tN0t−1 ⊆ N0}. Then for any locally analytic representation π

of B we have a Hecke action of T+ on πN0 by t ·v :=∑

N0/tN0t−1 ntv for t ∈ T+ and v ∈πN0 (cf. [Eme06a, §3.4], but note that we do not normalise). Let K :=

∏di=1 GLn(OLi).

Proposition 4.1 (Emerton). Suppose that Π is an admissible continuous represen-tation of G on a Banach space and that 0 → ΠQp-an → π → π1 → 0 is an exactsequence of admissible locally Qp-analytic representations of G, where ΠQp-an ⊆ Π isthe subspace of locally Qp-analytic vectors for the action of G (an admissible locally


Qp-analytic representation of G by [ST03, Thm. 7.1]). Suppose that χ : T → E× is a

locally Qp-analytic character and η : t→ E its derivative. If H1(b,ΠQp-an ⊗ (−η)) = 0then we have short exact sequences

0→ (ΠQp-an)N0 [t = η]→ πN0 [t = η]→ πN01 [t = η]→ 0,(4.2)

0→ (ΠQp-an)N0 [t = η]χ → πN0 [t = η]χ → πN01 [t = η]χ → 0,(4.3)

where Vχ ⊆ V denotes the generalised χ-eigenspace under the action of T+, and allvector spaces in the last exact sequence are finite dimensional.

Proof. This is the statement of [Brea, Prop. 6.3.3], and we (finally!) give here fulldetails of its proof. Let T 0 := T ∩K. The exactness of (4.2) follows as in the proof ofloc.cit. from H1(b,ΠQp-an ⊗ (−η)) = 0. To see the finite-dimensionality, we first showthat

(4.4) πN0 [t = η]χ = JB(π)[t = η]χ,

where JB denotes Emerton’s locally analytic Jacquet functor for the subgroup B of G([Eme06a]). Note that the T+-action on the left extends uniquely to a T -action. Wehave

(4.5) πN0 [t = η][(t− χ(t))s : t ∈ T+] = JB(π)[t = η][(t− χ(t))s : t ∈ T+]

for all s ≥ 1 by an analogue of [Eme06a, Prop. 3.2.12], noting that the T -action onthe left is locally Qp-analytic and T -finite (as T 0 acts semisimply and T/T 0 is finitelygenerated). Taking the union of (4.5) over all s ≥ 1, we obtain (4.4), since again T 0

acts semisimply and T/T 0 is finitely generated.Next, [Eme06a, Prop. 4.2.33] shows that the strong dual JB(π)[t = η]′b is the space

of global sections of a coherent sheaf F on T (the rigid analytic variety of locally Qp-analytic characters of T ) with discrete support. We write T =

⋃n≥1 Tn as increasing

union of an admissible cover of affinoid subdomains. Then F(Tn) ∼= ⊕x∈TnF(Tn)mx ,

where the direct sum is over all closed points x of Tn and mx is the corresponding

maximal ideal. It is a finite decomposition, as suppF is discrete; moreover, ms(x)x

annihilates F(Tn)mx for some s(x) ≥ 1 and F(Tn) is finite-dimensional. Also, F(Tn)mx∼=

Fx by [BGR84, Prop. 9.4.2/6]. By passing to the limit over n we get a topologicalisomorphism JB(π)[t = η]′b

∼=∏x∈T Fx and dually

JB(π)[t = η] ∼=⊕x∈T

(Fx)′b.

For any t ∈ T the function t−χ(t) on T is contained in the maximal ideal mχ of χ ∈ T ,

hence by above (t − χ(t))s(χ) annihilates Fχ for all t ∈ T . Therefore, JB(π)[t = η]χ ∼=(Fχ)′b is finite-dimensional.

To see the exactness of (4.3), we first note that by the above we can identify it withthe sequence 0→ JB(ΠQp-an)[t = η]χ → JB(π)[t = η]χ → JB(π1)[t = η]χ → 0, and thus

it is independent of the choice of N0.

Next, let H :=∏di=1{g ∈ GLn(OLi) : g ≡ 1 (mod pr)} for some large r and N0 :=

H ∩N (these are uniform pro-p groups). Then T 0 normalises N0, so the Hecke action

of T 0 on (ΠQp-an)N0 coincides with its natural action. Using [Eme06a, Prop. 3.3.2] we


choose z1, . . . , zs ∈ T+ such that ziN0z−1i ⊆ N

p0 for all i and such that T 0, z1, . . . , zs

generate T as group. By increasing r we may assume that moreover z−1i Hzi ⊆ K for

all i. Then by [BHS17b, Lemma 5.3] we know that ΠQp-an is an increasing union of

H-stable BH-subspaces Π(h) for h ≥ 1 and, moreover, for each h ≥ 1 the elements

zi preserve the BH-subspace Πh := Π(h) ∩ (ΠQp-an)N0 [T 0 = χ] and induce a compact

operator on the Banach space Πh (a closed subspace of Π(h) ∩ (ΠQp-an)N0 [t = η]), see§1.1 for the notation.

We observe that if Y ⊆ T+ is any submonoid that contains T 0 and generates T asgroup, then we can also compute JB(π) using Y instead of T+ by [Eme06a, Lemma3.2.19]. In fact, all results of [Eme06a, §3.2] apply and we deduce in particular that

πN0 [t = η]Y=χ = JB(π)[t = η]Y=χ by the argument at the beginning of our proof,where the subscript Y = χ denotes the generalised eigenspace for χ|Y . As Y generates

T as group we also have JB(π)[t = η]Y=χ = JB(π)[t = η]χ. Thus the space πN0 [t =

η]Y=χ = πN0 [T 0 = χ]Y=χ is finite-dimensional and independent of the choice of Y .Let Y denote the submonoid of T+ generated by T 0, z1, . . . , zs. From (4.2) we deduce

an exact sequence 0 → (ΠQp-an)N0 [T 0 = χ] → πN0 [T 0 = χ] → πN01 [T 0 = χ] → 0

and it remains to show that the last map is surjective on generalised χ|Y -eigenspaces.

Let M1 := πN01 [T 0 = χ]Y=χ = πN0

1 [t = η]χ and let M be the preimage of M1 in

πN0 [T 0 = χ]. Let V be any subspace of πN0 [T 0 = χ] mapping isomorphically ontothe finite-dimensional space M1. Now choose h ≥ 1 such that ziV ⊆ V + Πh for all

i (recall (ΠQp-an)N0 [T 0 = χ] is the increasing union of the Πh), and let V ′ := Πh ⊕ V(a BH-subspace of πN0 [T 0 = χ]). We have a Y -equivariant exact sequence of Banachspaces 0→ Πh → V ′ →M1 → 0. As the zi are compact on Πh and dimEM1 <∞, wededuce that the zi are compact on V ′. It follows that the generalised eigenspace V ′Y=χ

is finite-dimensional and surjects onto M1. This finishes the proof of (4.3). �

Example 4.6. Suppose that Π|K ∼= C0(K,E)⊕r for some r, or just that ΠQp-an|K ∼=CQp-an(K,E)⊕r, then H1(b,ΠQp-an ⊗ (−η)) = 0 for all linear maps η : t → E by theproof of [Brea, Prop. 6.3.3].

We will need the following corollary for future reference.

Corollary 4.7 (Emerton). Suppose that π is an admissible locally Qp-analytic rep-resentations of G and that χ : T → E× is a locally Qp-analytic character. Then

HomT+(χ, πN0) = HomT (χ, JB(π)) is finite-dimensional.

Proof. The two spaces are equal by [Eme06a, Prop. 3.2.12]. Moreover, the image of

any T+-equivariant linear map χ → πN0 has to be contained in πN0 [t = η]χ, which isfinite-dimensional by Proposition 4.1. �

Proposition 4.8. Suppose that Π is an admissible continuous representation of G ona Banach space together with an action of a commutative algebra T by continuous G-linear endomorphisms. Let m C T be a maximal ideal with residue field E and σ anadmissible locally Qp-analytic representation of G of finite length such that:

(i) we have H1(b,ΠQp-an ⊗ (−η)) = 0 for all linear maps η : t→ E;(ii) we are given a (G-equivariant continuous) injection j : socG σ ↪→ ΠQp-an[m];


(iii) any irreducible constituent of σ/ socG σ is of the form FGB(L(µ), χ) for someµ ∈ X∗(T ) and some smooth character χ : T → E× such that moreover noirreducible subquotient of FGB(M(µ), χ) injects into ΠQp-an[m].

Then j extends uniquely to a (G-equivariant continuous) map σ → ΠQp-an[m] that ismoreover injective.

Proof. The uniqueness of the extension is clear by the last assumption. To showexistence we follow the strategy of [Brea, §6.4]. For existence, we may assume byinduction that we have already extended j to an injection j0 : σ′ ↪→ ΠQp-an[m],where socG σ ⊆ σ′ ⊆ σ and C := σ/σ′ is irreducible. We define the amalgamS := ΠQp-an ⊕j0,σ′ σ and let T act on S by declaring that it acts through T/m ∼= E onσ. Then S[m] = ΠQp-an[m]⊕j0,σ′ σ, and so we have an exact sequence 0→ ΠQp-an[m]→S[m] → C → 0. By assumption, we can write C ∼= FGB(L(µ), χ), and hence by [Bre15,Thm. 4.3],

(4.9) HomT+(t−µχ,CN0) 6= 0.

(We note that [Bre15] works with the group of L-points of a split reductive group overL. However, the proofs work unchanged for our group G. See also [BHS17a, Rk. 5.1.2].)Let η : t → E denote the derivative of t−µχ, i.e. η = −µ. By Proposition 4.1 we havean exact sequence of finite-dimensional vector spaces

(4.10) 0→ (ΠQp-an)N0 [t = η]t−µχ → SN0 [t = η]t−µχ → CN0 [t = η]t−µχ → 0.

If we had ((ΠQp-an)N0 [t = η]t−µχ)m 6= 0 then HomT+(t−µχ, (ΠQp-an[m])N0) 6= 0, so by

[Bre15, Thm. 4.3] we would get a non-zero G-linear map FGB(M(µ)∨, χ) → ΠQp-an[m]

(see loc.cit. for the notation) and hence some irreducible constituent of FGB(M(µ)∨, χ),or equivalently of FGB(M(µ), χ), would inject into ΠQp-an[m], contradicting our lastassumption. Therefore the second and third terms in the sequence (4.10) becomeisomorphic after m-localisation, and thus also on taking m-torsion. In particular, wehave an isomorphism

(4.11) HomT+(t−µχ, S[m]N0)∼−→ HomT+(t−µχ,CN0),

and they are non-zero by (4.9). (In fact they are one-dimensional, but we will not needthat.)

As in [Brea, §6.4] a non-zero element of HomT+(t−µχ, S[m]N0) corresponds to anon-zero (g, B)-linear map

(4.12) U(g)⊗U(b) (−µ)⊗E C∞c (N,χ)→ S[m],

where C∞c (N,χ) is the B-representation defined in loc.cit. twisted by the character χ.The map (4.12) factors through a map

ψ1 : C lp(N) = L(−µ)⊗E C∞c (N,χ)→ S[m]

by our last assumption. (For C lp(N) ⊆ C we use the notation of [Eme, §2.7] and forthe identification with L(−µ) ⊗E C∞c (N,χ), see the proof of [Bre15, Prop. 4.2].) Bythe isomorphism (4.11) we may assume that the composite of ψ1 with the natural mapS[m]→ C is the inclusion C lp(N)→ C.


Let us write σ = (σ′ — C) and let σ′ — C lp(N) be the pullback of σ along theinclusion C lp(N) ↪→ C. Let ψ2 be the composite σ′ — C lp(N) ↪→ σ ↪→ S[m], so ψ2|σ′ =j0 and let s denote the projection (σ′ — C lp(N)) � C lp(N). Then by construction,ψ := ψ2 − ψ1 ◦ s : (σ′ — C lp(N)) → ΠQp-an[m] and it restricts to j0 on σ′. By [Brea,Thm. 7.1.1], the map ψ extends uniquely to a G-equivariant map σ → ΠQp-an[m]. Thislatter map is injective, since it is non-zero on socG σ (as j0 is). �

Remark 4.13. At least in our global application, the condition that no irreduciblesubquotient of FGB(M(µ), χ) injects into ΠQp-an[m] in Proposition 4.8(iii) will be equiv-

alent to demanding that FGB(L(µ), χ) does not inject into ΠQp-an[m] by Conjecture 5.10(which holds in many cases by [BHS17a, Thm. 1.3]), see the proof of Theorem 5.12.

Remark 4.14. We note that Proposition 4.8 generalises [BC18, Thm. B] when theparabolic subgroup P of loc.cit. is the Borel subgroup and the character is locallyalgebraic.

5. The finite slope space in the generic crystabelline case

We prove our main result (Corollary 5.16).

5.1. Local setup and results. We define and study the “finite slope” representationΠ(ρ)fs.

Let L/Qp be a finite extension and set SL := HomQp(L,E). We fix a crystabelline

representation ρ : Gal(L/L)→ GLn(E) satisfying the following genericity hypothesis.

Hypothesis 5.1. We assume that ρ is potentially crystalline with WD(ρ) = ⊕ni=1χifor some smooth characters χi : WL → E× (and N = 0), that χiχ

−1j 6∈ {1, | · |

±1L } for

all i 6= j, and that for each σ ∈ SL the σ-Hodge–Tate weights of ρ are distinct.

Let L′/L be finite abelian such that ρ|Gal(L/L′) is crystalline. Then, in particular,

the Deligne–Fontaine module D := Dcris(ρ|Gal(L/L′)) covariantly associated to ρ satisfies

Hypotheses 5.1 and 5.2 in [Bre16]. For each σ ∈ SL, let h1,σ < · · · < hn,σ denote thejumps in the Hodge filtration on DL′,σ := DL′ ⊗L⊗QpE,σ E (or equivalently, in the n-

dimensional E-vector space DGal(L′/L)L′,σ ), and let λi,σ := −hi,σ− (n− i) for all 1 ≤ i ≤ n,

so that λ1,σ ≥ · · · ≥ λn,σ.

Let G := GLn/L and G := ResL/Qp G×Qp E ∼=∏

SLGLn/E . Let B (resp. B) denote

the lower-triangular (resp. upper-triangular) Borel subgroup of G, let T denote thediagonal maximal torus and W ∼= Sn the Weyl group of (G,T ) with Bruhat order ≤,and let w0 denote the longest element of W . We let ∆ ⊆ X(T ) denote the simple rootsof G with respect to B. Similarly we define B,B, T ,W,w0,∆ for G. Note that we cannaturally identify W as a subgroup of W . Also, we can and will think of λ = (λi,σ)i,σas an element of X(T ), which is dominant with respect to B.

We say that a refinement of D is a complete flag of Deligne–Fontaine submodules ofD. Equivalently, it is a complete flag of Weil–Deligne subrepresentations of WD(D) =WD(ρ) ∼= ⊕ni=1χi. Since the characters χi are distinct by assumption, every refinementof D is of the form

(5.2) F : 0 ⊆ DF,1 ⊆ DF,1 ⊕DF,2 ⊆ · · · ⊆ D,


where the DF,i are pairwise distinct Deligne–Fontaine submodules of D of rank 1. Let

χF,i := WD(DF,i), which is a smooth character of W abL or equivalently of L×. Thus we

see that a refinement of D can also be thought of as an ordering χF,1, . . . , χF,n of thecharacters χi. Moreover, we have a simply transitive action of W on the set Ref(D) ofrefinements of D, given by

wF : 0 ⊆ DF,w−1(1) ⊆ DF,w−1(1) ⊕DF,w−1(2) ⊆ · · · ⊆ D,where w ∈W and F ∈ Ref(D) is as in (5.2), i.e. χwF,i = χF,w−1(i).

Let W := W ×Ref(D). For (walg,F) ∈W we define a locally Qp-algebraic character

η(walg,F) : T (L)→ E× as follows:

(5.3)

( x1

. . .xn

)∈ T (L) 7−→

n∏i=1

( ∏σ∈SL

σ(xi)−h

(walg)−1(i),σ

)χF,i(xi)ε

−(n−i)(xi)

and we let πB,F := χF,1| · |−(n−1)L ⊗ χF,2| · |

−(n−2)L ⊗ · · · ⊗ χF,n denote the smooth part

of η(walg,F). We define the locally Qp-analytic principal series representation of G(L)

(5.4) PS(walg,F) :=(

IndG(L)B(L) η(walg,F)

)Qp-an

(which is of compact type and admissible). For walg ∈ W let W (walg) denote thesubgroup of W generated by all reflections sα for α ∈ ∆ with sαw

alg > walg (where wesee W as subgroup of W ) and P (walg) the parabolic subgroup of G containing B withWeyl group W (walg).

Lemma 5.5.

(i) The principal series PS(walg,F) has finite length and irreducible socle, whichwe denote by C(walg,F).

(ii) We have C(walg1 ,F1) ∼= C(walg

2 ,F2) if and only if walg1 = walg

2 and W (walg1 )F1 =

W (walg2 )F2.

(iii) The irreducible constituents of PS(walg,F) are the representations C(τ,F) forτ ∈W , τ ≥ walg. They occur only once in case τ ∈ {walg, w0}.

Proof. Part (i) is a special case of the results in [Bre16, §6]: using the theory ofOrlik–Strauch [OS15], as extended in [Bre16], we can write PS(walg,F) = FGB(M(walg ·(−λ)), πB,F), so by [Bre16, Cor. 2.5] we have that C(walg, w) = FGB(L(walg ·(−λ)), πB,F)

is the (irreducible) socle of PS(walg,F). (We remark that for us, the construction ofOrlik–Strauch, as well as the dot action of W on X(T ), are defined relative to our choiceof lower-triangular Borel subgroup B, just as in [Bre15]. In [Bre16] the dot action wasdefined relative to B.) Part (ii) now follows from [Bre16, Lem. 6.2], the sentence before[Bre16, Lem. 6.3], and [Bre16, Lem. 4.2]. Since −λ is dominant with respect to B,the Verma module M(walg · (−λ)) has finite length and constituents L(τ · (−λ)) forτ ≥ walg, which occur only once in case τ ∈ {walg, w0} (see for example the bottomof p. 155 in [Hum08] for the latter case). As the smooth induction (IndGB πB,F)∞ isirreducible by genericity, parts (i) and (iii) follow from the main results of [OS15]. �

By base change a refinement F ∈ Ref(D) gives rise to a Gal(L′/L)-stable flag FL′ ofDL′ := L′ ⊗L′0 D. On the other hand, forgetting the indexation, the Hodge filtration

Fil∗DL′ gives rise to another Gal(L′/L)-stable flag of DL′ . By Galois descent we obtain


two complete flags (FL′)Gal(L′/L), (Fil∗DL′)

Gal(L′/L) on the free rank-n L⊗QpE-module

(DL′)Gal(L′/L). Their relative position is given by an element of W , which we denote by

walg(F)w0 (thus defining walg(F)). Explicitly, if α : (L⊗QpE)n∼−→ (DL′)

Gal(L′/L) is any

isomorphism of L⊗QpE-modules, the flags α−1((FL′)Gal(L′/L)), α−1((Fil∗DL′)

Gal(L′/L))are described by an element of

G(E)\((G(E)/B(E))× (G(E)/B(E))

)which is independent of the choice of α. We mean that this element is in the samecoset as (1, walg(F)w0).

Remark 5.6. We relate C(walg,F) and walg(F) to the notions introduced in [Bre16,§6], [Bre15, §6]. In those references, a refinement F is fixed at the outset to defineC(walg, w) and walg(w). To indicate the dependence on F, we write CF(walg, w) and

walgF (w) in this remark. With this convention, we have CF(walg, w) = C(walg, wF) and

walgF (w) = walg(wF). (The latter equality follows from walg

F (w) = walgwF(1), which holds

by an elementary argument just as in the proof of [Bre16, Prop. 6.4(i)].)

Now, we let Wsoc(ρ) := {(walg,F) ∈W : walg ≤ walg(F)} and

Csoc(ρ) := {C(walg,F) (up to isomorphism) such that(walg,F) ∈Wsoc(ρ)}.

We recall that if C(walg1 ,F1) ∼= C(walg

2 ,F2), then walg1 = walg

2 =: walg and we havewalg ≤ walg(F1) if and only if walg ≤ walg(F2) (as follows from [Bre15, Lem. 6.3]).The following construction takes place in the abelian category of admissible locallyQp-analytic representations (in fact, all representations that are involved are of finitelength).

Definition 5.7.

(i) For any (walg,F) ∈ Wsoc(ρ), let M(walg,F) be the largest (non-zero) subrep-resentation of the principal series PS(walg,F) such that none of the irreducibleconstituents of M(walg,F)/C(walg,F) is contained in Csoc(ρ).

(ii) For any C ∈ Csoc(ρ) let M(ρ)C denote the amalgam⊕C

{M(walg,F) : (walg,F) ∈Wsoc(ρ), C(walg,F) ∼= C}

over the common socle C.(iii) Let Π(ρ)fs

C denote the unique quotient of M(ρ)C whose socle is isomorphic toC.

(iv) Let Π(ρ)fs :=⊕

C∈Csoc(ρ) Π(ρ)fsC .

Remark 5.8.

(i) By construction, C is contained in the socle of M(ρ)C , but equality does nothold in general (see the examples in §5.3). Moreover, the quotient Π(ρ)fs

C iswell-defined, since C occurs just once in M(ρ)C .

(ii) Note that M(walg,F) = PS(walg,F) if and only if walg = walg(F) (use Lemma5.5(iii)).

(iii) Note that M(walg,F) injects into Π(ρ)fsC(walg,F)

for each (walg,F) ∈Wsoc(ρ).

Proposition 5.9. Suppose that F ∈ Ref(D).


(i) If α ∈ ∆, then C(sα,F) occurs precisely once in Π(ρ)fs.(ii) The representation C(w0,F) occurs precisely once in Π(ρ)fs.

Proof. (i) If (sα,F) ∈ Wsoc(ρ), then C := C(sα,F) is a subrepresentation of Π(ρ)fs

and occurs only once. Otherwise, if C occurs in M(walg,F′) with (walg,F′) ∈Wsoc(ρ),then walg = 1 and W (sα)F = W (sα)F′ by Lemma 5.5. By the Kazhdan–Lusztigconjectures (or Jantzen’s multiplicity 1 criterion), the Verma module M(−λ) containsthe constituent L(sα · (−λ)) with multiplicity one, and in the second radical layer.Therefore PS(1,F′), having an irreducible socle, contains a non-split extension E ofthe form C(1, 1)—C as subrepresentation, and hence so does M(1,F′). More precisely,we have E ∼= FGB(M,πB,F′), where M is the length two quotient of M(−λ) with socleL(sα · (−λ)). As sα · (−λ) and −λ are dominant with respect to the Borel B∩LP (sα) of

LP (sα), it follows that E ∼= FGP (sα)(M, (IndP (sα)(L)B(L) πB,F′)

∞), and hence is independent

of the choice of F′, as (IndP (sα)(L)B(L) πB,F′)

∞ is by the genericity conditions. Therefore,

all occurrences of C inside M(ρ)C(1,1) are contained in an amalgam of r := #W (sα)copies of the extension E over the common socle C(1, 1). But this amalgam is easilyseen to be isomorphic to E⊕ Cr−1, hence there is only one copy of C in Π(ρ)fs

C(1,1).

(ii) On the one hand, M(walg(F),F) = PS(walg(F),F) contains C := C(w0,F)precisely once as constituent by Lemma 5.5. On the other hand, if C occurs inM(walg,F′) with (walg,F′) ∈ Wsoc(ρ), then F = F′ as W (w0) = 1, so walg ≤ walg(F).If walg < walg(F), then we have a surjection PS(walg,F) � PS(walg(F),F) (becauseof a corresponding injection of Verma modules). It has to send M(walg,F) to zero, asotherwise M(walg,F) would contain C(walg(F),F) ∈ Csoc(ρ) as constituent (not in itssocle), so C does not occur in M(walg,F). Hence walg = walg(F) and we are done, sinceM(walg(F),F) injects into Π(ρ)fs

C(walg(F),F). �

5.2. Global applications. We prove our main global results (Theorem 5.12 andCorollary 5.16).

We first explain our global setup, which is essentially the same as that of [Bre15,§5–§6] (except we do not assume that p splits completely in our totally real field),to which we refer the reader for further details and references. We fix a totally realnumber field F+ 6= Q and a totally imaginary quadratic extension F/F+. We let cdenote the unique complex conjugation of F and suppose that every place v|p of F+

splits in F . We let G be a unitary group over F+ defined by a hermitian form of rank nover F , so we have an isomorphism ιG : G×F+ F

∼−→ GLn. We assume moreover thatthe hermitian form is totally definite, i.e. G(F+ ⊗Q R) is compact. We fix a compactopen subgroup Up of G(A∞,p

F+ ) of the form Up =∏v-p Uv for compact open subgroups

Uv ⊆ G(F+v ) for v - p. We define

S(Up, E) := C0(G(F+)\G(A∞F+)/Up, E),

which is a Banach space for the supremum norm on the (profinite) compact topological

space G(F+)\G(A∞F+)/Up. We will sometimes just write S for S(Up, E). For any place

v of F+ that splits as v = wwc in F , we choose an isomorphism ιw : G(F+v )

∼−→GLn(Fw) that is conjugate to the isomorphism induced by ιG on Fw-points (the choiceof which won’t matter). If in addition Uv is a maximal compact subgroup of G(F+

v ),


then we demand moreover that ιw(Uv) = GLn(OFw). We let Σ(Up) denote the (finite)set of places v of F+ that split in F and are such that Uv is not maximal compact.

We let T(Up) = E[T(j)w ] denote the polynomial algebra over E generated by all T

(j)w for

w a place of F lying over a place v of F+ that splits in F and such that v 6∈ Σ(Up)

and v - p. Then T(Up) acts topologically on S(Up, E) by letting T(j)w act as the double

coset operator [Uv ι−1w

( 1n−j$w1j

)Uv], where $w is a uniformiser of Fw. This action

commutes with the unitary left action of G(F+p ) on S(Up, E) by right translation of

functions, where F+p := F+ ⊗Q Qp. For each place v|p of F+ we choose a place v|v of

F (the choice of which won’t matter). We note that F+p∼=∏v|p Fv and that G(F+

p ) is

identified with GLn(Fv) via ιv.We let r : Gal(F/F )→ GLn(E) be a continuous representation and we assume:

(i) r is unramified at the places of F+ that split in F and are not in Σ(Up);(ii) rc ∼= r∨ ⊗ ε1−n (where rc(g) := r(cgc), g ∈ Gal(F/F ) and r∨ is the dual of r);

(iii) r is an absolutely irreducible representation of Gal(F/F ).

We associate to r and Up the maximal ideal mr in T(Up) generated by all elements((−1)jNorm(w)j(j−1)/2T (j)

w − a(j)w

)j,w,

where j ∈ {1, . . . , n}, w is a place of F lying over a place v of F+ that splits in F andsuch that v 6∈ Σ(Up) and v - p, Norm(w) is the cardinality of the residue field at w,

and where Xn + a(1)w Xn−1 + · · · + a

(n−1)w X + a

(n)w is the characteristic polynomial of

r(Frobw) (an element of O[X], Frobw is a geometric Frobenius element at w). We as-

sume that S(Up, E)Qp-an[mr] has non-zero locally Qp-algebraic vectors (where (−)Qp-an

is the locally convex subspace of locally Qp-analytic vectors for the action of G(F+p )),

i.e. r is automorphic of “level” Up.We now assume in the following that for each place v|p of F+ the representation

rv := r|Gal(Fv/Fv) is crystabelline and satisfies Hypothesis 5.1. We use the notation of

§5.1 modified in a trivial way as follows: we let W :=∏v|pWv, W

soc(r) :=∏v|pW

socv (rv)

and we write Cv(walgv ,Fv), Mv(w

algv ,Fv), etc. for the representations of GLn(Fv) defined

in §5.1. Usually we will omit the subscript v on the outside, which should not lead toany confusion.

Recall the following conjecture of the first author (slightly generalised, since we donot assume that p splits completely in F+).

Conjecture 5.10. [Bre15, Conj. 6.1] Suppose that (walgv ,Fv)v ∈W. We have

HomG(F+p )(⊗

v|pC(walg

v ,Fv)(εn−1), S(Up, E)Qp-an[mr]) 6= 0

if and only if (walgv ,Fv)v ∈Wsoc(r) (where ∗(χ) := ∗ ⊗ (χ ◦ det)).

Recall this is known to be true in many cases where rv is crystalline for all v|p.

Theorem 5.11. [BHS17a, Thm. 1.3] Assume that Up is sufficiently small ([CHT08,§3.3]), the residual representation r is absolutely irreducible, rv is crystalline for all v|p(and satisfies Hypothesis 5.1), and that the following assumptions hold:

(i) p > 2;


(ii) F/F+ is unramified and G is quasi-split at all finite places of F+;(iii) Uv is hyperspecial when the finite place v of F+ is inert in F ;(iv) r(Gal(F/F ( p

√1)) is adequate ([Tho12, Def. 2.3]).

Then Conjecture 5.10 is true.

Before stating our theorem, we observe that for any (walgv ,Fv)v ∈ W the represen-

tation ⊗v|pC(walg

v ,Fv)(εn−1) of G(F+

p ) ∼=∏v|p GLn(Fv) is admissible and topologically

irreducible. This follows from Lemma 2.10 and the theory of Orlik–Strauch. Below wewill tacitly use the exactness of ⊗ for compact type spaces (Corollary 2.2) and thatthe external tensor product of admissible locally analytic representations is admissible(Lemma 2.20).

Theorem 5.12. Suppose that Conjecture 5.10 holds. If Cv ∈ Csoc(rv) for all v|p,

then any injective G(F+p )-equivariant homomorphism ⊗

v|pCv(ε

n−1)→ S(Up, E)Qp-an[mr]

extends uniquely to a G(F+p )-equivariant homomorphism

⊗v|p

Π(rv)fsCv

(εn−1) −→ S(Up, E)Qp-an[mr]

that is moreover injective.

Proof. We let f denote the given injection ⊗v|pCv(ε

n−1) ↪→ S(Up, E)Qp-an[mr] and we

use the notation of §5.1.Step 1: We show that f extends uniquely to a G(F+

p )-equivariant homomorphism

(5.13) ⊗v|pM(rv)Cv(ε

n−1) −→ S(Up, E)Qp-an[mr].

By construction (see Definition 5.7(ii)), and by Lemma 2.3, it suffices to show that

for any fixed (walgv ,Fv)v ∈ Wsoc(r) such that Cv ∼= C(walg

v ,Fv) for all v|p the map fextends uniquely to a G(F+

p )-equivariant homomorphism

⊗v|pM(walg

v ,Fv)(εn−1) −→ S(Up, E)Qp-an[mr].

This follows by applying Proposition 4.8 for the group G(F+p ), Π = S(Up, E), σ =

⊗v|pM(walg

v ,Fv)(εn−1), T = T(Up), m = mr, as we now explain. Note first that

⊗v|pM(walg

v ,Fv) is of finite length (by the remark preceding the theorem). As Cv only

occurs once in M(walgv ,Fv) for all v|p, it easily follows from Lemma 2.21 that the

G(F+p )-socle of ⊗v|pM(walg

v ,Fv) is isomorphic to ⊗v|pCv. Any irreducible constituent of

⊗v|pM(walg

v ,Fv)/ ⊗v|pCv has the form ⊗

v|pC(τv,Fv), where τv ≥ walg

v and for at least one

v′|p we have τv′ � walgv′ (Fv′). Using Lemma 2.10 we have ⊗

v|pC(τv,Fv) ∼= F

G(F+p )

B(F+p )

(L(µ), χ)

with µ = (τv ·(−λv))v|p and χ =∏v|p πBv ,Fv . By Lemma 2.10 again, F

G(F+p )

B(F+p )

(M(µ), χ) ∼=⊗v|p

PS(τv,Fv) which has irreducible constituents ⊗v|pC(τ ′v,Fv) with τ ′v ≥ τv for all v|p, so


τ ′v′ � walgv′ (Fv′). Therefore, by Conjecture 5.10, we may indeed apply Proposition 4.8

and the claim follows.Step 2: We show that the map (5.13) factors uniquely through a map

(5.14) ⊗v|p

Π(rv)fsCv

(εn−1) −→ S(Up, E)Qp-an[mr]

which is moreover injective. By Corollary 2.2 we have a surjection ⊗v|pM(rv)Cv �

⊗v|p

Π(rv)fsCv

with kernel∑

v|pKv ⊗( ⊗v′ 6=v

M(rv′)Cv′ ), where Kv := ker(M(rv)Cv �

Π(rv)fsCv

). By definition, no irreducible constituent of Kv is contained in Csoc(rv),

hence the claim follows from Conjecture 5.10. The resulting map (5.14) is injective,since it is non-zero, the left-hand side has socle ⊗

v|pCv by Lemma 2.21, and no other

irreducible constituent injects into the right-hand side (by Conjecture 5.10). �

Lemma 5.15. For (walgv ,Fv)v ∈W the E-vector space

HomG(F+p )

(⊗v|pC(walg

v ,Fv)(εn−1), S(Up, E)Qp-an[mr]

)is finite-dimensional.

Proof. Let Π := S(Up, E)Qp-an[mr], which is an admissible locally Qp-analytic G(F+p )-

representation. As in (4.9) and using the notation of §4 (recallG(F+p ) ∼=

∏v|p GLn(Fv)),

we can find a locally Qp-analytic character χ : T → E× and a non-zero T+-equivariant

homomorphism f : χ → CN0 , where C := ⊗v|pC(walg

v ,Fv). Then restriction to N0-

invariants and composition with f gives a map

HomG(F+p )

(⊗v|pC(walg

v ,Fv)(εn−1),Π

)−→ HomT+(θ,ΠN0),

(where θ is the relevant twist of χ) which is injective, as ⊗v|pC(walg

v ,Fv) is irreducible.

The latter space is finite-dimensional by Corollary 4.7. �

Corollary 5.16. We keep the hypotheses of Theorem 5.12. For each C = ⊗v|pCv withCv ∈ Csoc(rv) let

nC := dimE HomG(F+p )

(⊗v|pCv(εn−1), S(Up, E)Qp-an[mr]

)∈ Z>0.

Then there exists an injective G(F+p )-equivariant linear map

(5.17)⊕

C=⊗Cv

(⊗v|p

Π(rv)fsCv

(εn−1))⊕nC −→ S(Up, E)Qp-an[mr].

Proof. By assumption we have an injection

(5.18)⊕

C=⊗Cv

(⊗v|pCv(ε

n−1))⊕nC ↪→ S(Up, E)Qp-an[mr].

Applying Theorem 5.12 to each irreducible direct summand, we see that the given mapextends uniquely to a map as in (5.17). The resulting map is injective because from(5.18) it is injective on the socle. �


Combining Corollary 5.16 with Theorem 5.11, we obtain the result in the introduc-tion.

5.3. Special cases and examples. We give explicit examples for the representationsΠ(ρ)fs and also relate Π(ρ)fs to previous results or conjectures. For simplicity, we onlyconsider here crystalline representations.

We first give two examples in the crystalline case for GL3(Qp). A refinement is herean ordering of the (distinct) eigenvalues {ϕ1, ϕ2, ϕ3} of the crystalline Frobenius. Wedenote by sα, sβ the two simple reflections, which generate the Weyl group W = S3.

We start with the noncritical case, by which we mean walg(F) = 1 for all refinementsF. We fix an arbitrary refinement F0 := (ϕ1, ϕ2, ϕ3) (the choice of which won’t matter)and recall that wF0 = (ϕw−1(1), ϕw−1(2), ϕw−1(3)) for w ∈ S3. One can then check

that Π(ρ)fs has the following explicit form, where the constituent C(walg, wF0) is justdenoted Cwalg,w below, where the (irreducible) socle is the constituent C(1,F0) = C1,1

in the middle, where we use without comment the intertwinings provided by Lemma5.5(ii), where a line between two constituents means as usual a non-split extension assubquotient and where the constituent further away from the centre is the quotient:

C1,1

Csα,1

Csβ ,1

Csα,sα

Csβ ,sβsα

Csβ ,sβ

Csα,sαsβ

Csαsβ ,1

Csβsα,1

Csαsβsα,1

Csαsβ ,1

Csβsα,sβ

Csαsβsα,sβ

Csβsα,1

Csαsβ ,sα

Csαsβsα,sα

Csβsα,sβsα

Csαsβ ,sα

Csαsβsα,sβsα

Csβsα,sβsα

Csαsβ ,sαsβ

Csαsβsα,sαsβsα

Csαsβ ,sαsβ

Csβsα,sβ

Csαsβsα,sαsβ

(Note with Lemma 5.5(ii) that Π(ρ)fs is not multiplicity free: the 6 distinct constituentsof the form C(sαsβ,F) or C(sβsα,F) all appear with multiplicity 2.)


We go on with an example in the critical ordinary case. Here we have a canonicalrefinement F0 = Fρ (see the beginning of §6.1) due to the fact that ρ is upper triangularwith distinct Hodge–Tate weights. The possible locally analytic socles are worked outin [Breb, §6.2], we only give here Π(ρ)fs when its socle is C(1,Fρ) ⊕ C(sα, sαsβFρ) =C1,1 ⊕Csα,sαsβ (the interested reader can easily work out the other cases). We get thefollowing form (same notation as before, the socle in each summand being now on theleft)

C1,1

Csβ ,sβ

Csα,1

Csβsα,sβ

Csαsβ ,1

Csαsβsα,sβ

Csβ ,1

Csαsβ ,1

Csβsα,1

Csαsβsα,1

Csα,sα

Csβ ,sβsα

Csαsβ ,sα

Csβsα,1

Csαsβsα,sα

Csβsα,sβsα

Csαsβ ,sα

Csαsβsα,sβsα

⊕Csα,sαsβ

Csβsα,sβ

Csαsβ ,sαsβ

Csαsβsα,sαsβ

Csβsα,sβsα Csαsβsα,sαsβsα

One can check that Π(ρ)fs again fails to be multiplicity free (4 constituents appear withmultiplicity 2) and that (Π(ρ)ord)Qp-an (see Proposition 6.18) is the direct summand onthe left.

In the crystalline case for GL2(L), the representation Π(ρ)fs is easily checked to beexactly the representation Π(Dcris(ρ)) in [Bre16, §4(9)]. In particular, in this case The-orem 5.12 was already proven by Ding in the setting of the completed H1 of unitaryShimura curves (see [Din, Thm. 6.3.7]). The proof of loc.cit. however is different fromthat of Theorem 5.12 (e.g. it doesn’t use [Brea, Thm. 7.1.1]). Note that here Π(ρ)fs ismultiplicity free.

Finally, Theorem 5.12 (assuming Conjecture 5.10) together with Proposition 5.9(i)imply that any constituent of the form

(⊗

v′|p,v′ 6=vC(1,Fv′)(ε

n−1))⊗C(sαv ,Fv)(ε

n−1),


where v|p and sαv is a simple reflection in ResFv/Qp GLn×QpE ∼=∏

SFvGLn/E (see §5.1),

that does not inject into S(Up, E)Qp-an[mr] is such that there is a non-split extension(⊗

v′|p,v′ 6=vC(1,Fv′)(ε

n−1))⊗(C(1,Fv)—C(sαv ,Fv)

)(εn−1)

that does inject into S(Up, E)Qp-an[mr]. By [Brea, §3.3] and together with Theorem5.11, this gives further evidence to [Brea, Conj. 6.1.1] in the crystalline case (note thatin loc.cit. it is assumed that there is only one v dividing p in F+, in which case thefactor ⊗

v′|p,v′ 6=vC(1,Fv′)(ε

n−1) disappears).

6. Ordinary representations

For L = Qp and ρ crystabelline upper triangular satisfying Hypothesis 5.1 we prove

that the locally analytic vectors of the representation Π(ρ)ord of [BH15, §3.3] is asubrepresentation of Π(ρ)fs, and then deduce strong evidence to [BH15, Conj. 4.2.2] inthe crystalline case using Theorem 5.16 (and Theorem 5.11).

6.1. Local setup and results. For L = Qp and ρ crystabelline upper triangular

satisfying Hypothesis 5.1 we prove (among other results) that (Π(ρ)ord)Qp-an is a sub-

representation of Π(ρ)fs (Proposition 6.18) and that Π(ρ)ord is its universal unitarycompletion (Proposition 6.20).

We keep the notation of §5 and specialise to the case where L = Qp and ρ is crysta-

belline ordinary, that is ρ : Gal(Qp/Qp)→ GLn(E) is of the form

(6.1) ρ ∼

ψ1 ∗ . . . ∗

ψ2 . . . ∗. . .

...ψn

and satisfies Hypothesis 5.1. We write ψi(x) = x−hiχi(x) (for x ∈ Q×p ) with χi =WD(ψi) smooth and hi ∈ Z. We remark that Hypothesis 5.1 implies that ρ is genericin the sense of [BH15, Def. 3.3.1].

As ρ is regular de Rham we may assume without loss of generality that h1 < · · · <hn. Thus D := Dcris(ρ) has a canonical refinement Fρ with χFρ,i = χi for all i, and

for walg, w ∈ W we write η(walg, w) := η(walg, wFρ), PS(walg, w) := PS(walg, wFρ),

C(walg, w) := C(walg, wFρ), and walg(w) := walg(wFρ).We fix a representative homomorphism in the conjugacy class ρ that is a good conju-

gate in the sense of [BH15, Def. 3.2.4], and we will also denote it by ρ. This is possibleafter conjugating by a suitable element of B(E) by [BH15, Prop. 3.2.3]. We emphasisethat the following definition depends on our choice of good conjugate.

Definition 6.2. We let Wρ = {w ∈ W : wρw−1 is upper-triangular}, where w is arepresentative of w (this is the inverse of the subset WCρ defined in [BH15, §3.2] and isin general different from the subset denoted by Wρ in [BH15, (14)]). For each w ∈Wρ

we let Σw ⊆ W consist of all (commuting!) products sα1 · · · sαr with r ≥ 0, αi ∈ ∆pairwise orthogonal, and sαiw 6∈Wρ for all i.


Note that if σ ∈ Σw and σ′ ≤ σ, then σ′ ∈ Σw. (In fact, σ′ =∏i∈I sαi for a unique

subset I ⊆ {1, . . . , r}.) Note also that in this case we can uniquely write σ = σ′′σ′ with`(σ) = `(σ′′) + `(σ′). (Namely, σ′′ =

∏i 6∈I sαi .)

Lemma 6.3. For any w ∈Wρ, σ ∈ Σw we have walg(σw) = w.

Proof. For any 0 ≤ i ≤ n, let Fw,i denote the member of the flag wFρ that has rank

i. The element walg(w)w0 gives the relative position of the flags (wFρ)Gal(L′/Qp)L′ and

(Fil∗DL′)Gal(L′/Qp). A calculation shows that this means that the filtration Fil∗DL′ ∩

(Fw,i)L′ on (Fw,i)L′ (omitting Galois invariants for simplicity) jumps precisely at theintegers {hwalg(w)−1(j) : 1 ≤ j ≤ i}.

Next we claim that for w ∈ W the flag wFρ (with induced structures) consists ofweakly admissible subobjects if and only if w ∈Wρ. The first condition is equivalent tosaying that ρ has a filtration with subquotients ψw−1(i), 1 ≤ i ≤ n in this order (wherei = 1 corresponds to the subobject), i.e.

(6.4) ρ ∼

ψw−1(1) . . . ∗. . .

...ψw−1(n)

.

By [BH15, Prop. 3.2.3] we may assume, after further conjugation by B(E), that theright-hand side of (6.4) is a good conjugate. By [BH15, Prop. 3.2.6] it then followsthat w ∈Wρ. The converse is clear.

Let us now go back to our given w ∈ Wρ and σ ∈ Σw. We can write σ =∏r`=1 s`

with s` corresponding to the simple root εn` − εn`+1 and such that n` + 1 < n`+1 forall 1 ≤ ` < r. Assume first σ = 1. By the previous paragraphs each subobject Fw,i isweakly admissible and the Hodge filtration of Fw,i jumps at {hwalg(w)−1(j) : 1 ≤ j ≤ i},from which we easily deduce walg(w) = w by induction. For general σ ∈ Σw as above,note that Fσw,i = Fw,i for i 6∈ {n1, . . . , nr} and that Fσw,n` = Fs`w,n` , which is notweakly admissible by the previous paragraph, as s`w 6∈ Wρ. As moreover Fσw,i = Fw,ifor i = n` ± 1, the only possibility is that the Hodge filtration of Fσw,n` jumps at

{hwalg(w)−1(j) : 1 ≤ j ≤ n`}. Hence again walg(σw) = w. �

We recall the following result of Breuil–Emerton ([BE10, Thm. 2.2.2]).

Proposition 6.5. Suppose that n = 2, that k1 > k2 are integers, and that θ1, θ2 :Q×p → E× are smooth characters such that x 7→ θi(x)(x)ki is unitary for i = 1, 2. If

k1 = k2+1, we further assume that θ1|·|−1Qp 6= θ2. Then the universal unitary completion

π of

π :=(

IndG(Qp)B(Qp) θ2(−)k1ε−1 ⊗ θ1(−)k2

)Qp-an

(where (−)k means the character x ∈ Q×p 7→ xk) is an admissible representation that is

a non-split extension of(

IndG(Qp)B(Qp) θ2(−)k2ε−1 ⊗ θ1(−)k1

)C0

by(

IndG(Qp)B(Qp) θ1(−)k1ε−1 ⊗

θ2(−)k2)C0

, each of which is topologically irreducible. Moreover if θ1θ±12 6∈ {1, | · |±1

Qp},then the canonical map π → π is injective.

Proof. We first reduce by twisting by a power of the unitary character ε to the casewhere k1 = 1. Then the first result follows from [BE10, Thm. 2.2.2]: in their notation


we need to take χ2 = θ2(−)k1ε−1 smooth, χ1 = θ1(−)k2 , and k = k1 − k2 + 1. Notealso that [BE10] work with the upper-triangular Borel B.

For the second result, recall that π is a non-split extension of π′′ :=(Ind

G(Qp)B(Qp) θ2(−)k2ε−1⊗θ1(−)k1

)Qp-anby socG(Qp)

(Ind

G(Qp)B(Qp) θ1(−)k1ε−1⊗θ2(−)k2

)Qp-an,

both of which are irreducible, as θ1θ±12 6∈ {1, | · |±1

Qp} (see e.g. [Breb, §3.2]). If the map

π → π isn’t injective, it thus has to factor through the quotient π′′. From the definitionof universal unitary completions it would then follow that π ∼= (π′′) . However, we

know that (π′′)∼= (IndG(Qp)B(Qp) θ2(−)k2ε−1 ⊗ θ1(−)k1)C

0(for example by Proposition 3.1,

but see also the proof of [BE10, Thm. 2.2.2]). This contradiction shows that the mapπ → π is indeed injective. �

Proposition 6.6. For any w ∈ Wρ, σ ∈ Σw, the locally analytic principal seriesPS(w, σw) admits an admissible universal unitary completion PS(w, σw) that is iso-morphic to the representation Π(ρ)w−1(J) constructed in [BH15, §3.3], where J = {α ∈∆ : sα ∈ supp(σ)}. Moreover, the canonical map PS(w, σw)→ PS(w, σw) is injective.

Proof. Let PJ be the parabolic containing B determined by the subset J ⊆ ∆ ofthe statement, let MJ be the Levi subgroup of PJ that contains T and let NJ de-

note the unipotent radical of PJ . We let π :=(

IndMJ (Qp)(B∩MJ )(Qp) η(w, σw)

)Qp-an, so that

PS(w, σw) ∼=(

IndG(Qp)PJ (Qp) π

)Qp-an.

From Proposition 3.1 applied to the character η(w, σw) we deduce that there existsa BH-subspace π0 ⊆ π such that π =

∑m∈MJ (Qp)mπ0 and that π exists. Moreover

observe that σ lies in the Weyl group of MJ with respect to T . Hence the centralcharacter η(w, σw)|ZMJ (Qp) of π is equal to η(w,w)|ZMJ (Qp), which is unitary by (5.3),

as each ψi is. Applying Proposition 3.1 again, this time to π, we deduce that

PS(w, σw) ∼=(

IndG(Qp)PJ (Qp) π

)Qp-anhas universal unitary completion

(Ind

G(Qp)PJ (Qp) π

)C0

.

We now determine π explicitly. There exist integers 1 ≤ n1 < · · · < nr < n suchthat ni + 1 < ni+1 for all i, and J consists of the simple roots εni − εni+1. Then wecan identify MJ with GLr2×GLn−2r

1 , where the i-th factor of GL2 corresponds to thesimple root εni − εni+1. By Lemma 2.8, π is the external completed tensor product of

all ψw−1(j)ε−(n−j) for σ(j) = j and

(6.7)(

IndGL2(Qp)

B2(Qp)χw−1(ni+1)(−)

−hw−1(ni)ε−(n−ni) ⊗ χw−1(ni)(−)−hw−1(ni+1)ε−(n−ni−1)

)Qp-an

for 1 ≤ i ≤ r, where B2 is the lower triangular Borel of GL2. For any 1 ≤ i ≤ r,as w ∈ Wρ and sαiw 6∈ Wρ, we note that the homomorphism wρw−1 contains the

2 × 2-submatrix

(ψw−1(ni) ∗

ψw−1(ni+1)

)with ∗ 6= 0. As ρ is by assumption a good

conjugate, we see that the extension ∗ is in fact non-split. It is moreover de Rham, asρ is, hence −hw−1(ni) > −hw−1(ni+1).

By Lemma 3.4 (both parts), Proposition 6.5, and Hypothesis 5.1 (and what is above),

we deduce that π is the external completed tensor product of all ψw−1(j)ε−(n−j) for

σ(j) = j and of the unique non-split extension of(

IndGL2(Qp)B2(Qp) ψw−1(ni+1)ε

−(n−ni) ⊗

ψw−1(ni)ε−(n−ni−1)

)C0

by(

IndGL2(Qp)B2(Qp) ψw−1(ni)ε

−(n−ni) ⊗ ψw−1(ni+1)ε−(n−ni−1)

)C0

for

1 ≤ i ≤ r (cf. [BH15, Prop. B.2] for the uniqueness). This is isomorphic to the


representation Π(ρ)w−1(J) constructed in [BH15, §3.3] (see in particular the construc-tion in Step 2 of the proof of [BH15, Prop. 3.3.3]). By the second paragraph it follows

that PS(w, σw)∼= ( IndG(Qp)PJ (Qp) Π(ρ)w−1(J)

)C0 ∼= Π(ρ)w−1(J), as desired.

We note that in the setting of Lemma 3.4, it follows from its proof that if eachcanonical map σi → σi is injective, then so is the canonical map σ1 ⊗π · · · ⊗π σr →σ1 ⊗ · · · ⊗ σr. Similarly, in the context of Proposition 3.1, if σ → σ is injective, then

so is (IndGP σ)Qp-an → (IndGP σ)C0. By the injectivity assertion of Proposition 6.5 and

by construction we thus deduce that the canonical map PS(w, σw) → PS(w, σw) isinjective. �

For w ∈ Wρ and σ, σ′ ∈ Σw with σ ≤ σ′, by Proposition 6.6 and [BH15, §3.3] thereexists a G(Qp)-equivariant embedding PS(w, σw) ↪→ PS(w, σ′w) that is unique upto scalar. As in that reference we can fix a compatible system of injections iw,σ,σ′ :PS(w, σw) ↪→ PS(w, σ′w) (the choice of which won’t matter) and obtain that

(6.8) Π(ρ)Cρ,w−1∼= lim−→

σ∈Σw

PS(w, σw)in the notation of that reference (see Definition 6.2).

We will need the following lemmas below.

Lemma 6.9. Suppose that w ∈Wρ and σ ∈ Σw.

(i) We have `(σw) = `(σ) + `(w).(ii) If w ≤ τ ≤ σw, then τ = σ′w with σ′ ≤ σ (hence σ′ ∈ Σw).

Proof. Write σ = sα1 · · · sαr with αi ∈ ∆ pairwise orthogonal and sαiw 6∈ Wρ for alli. The latter condition implies that w−1(αi) > 0 for all i. Note that σ is of length r,sending each αi to −αi and preserving Φ+ − {α1, . . . , αr}, where Φ+ ⊆ X(T ) (resp.Φ− ⊆ X(T )) denotes the positive (resp. negative) roots of G = GLn with respect to B.It follows that σw and w send precisely the same elements of Φ+ to Φ−, except thatw−1(α1), . . . , w−1(αr) are sent to Φ− by σw and to Φ+ by w. This implies (i).

For (ii) we induct on the length of σ. If σ = 1 there is nothing to show. Trivially wehave sα1σ < σ, hence by (i) we have sα1σw < σw.

If τ−1(α1) ∈ Φ+, then τ < sα1τ . By the lifting property of Coxeter groups, from theprevious two inequalities and τ ≤ σw we deduce that τ ≤ sα1σw. By the inductionhypothesis applied to sα1σ we deduce the claim.

If, on the other hand, τ−1(α1) ∈ Φ−, then sα1τ < τ and w < sα1w by (i). Asw ≤ τ , the lifting property of Coxeter groups gives w ≤ sα1τ . On the other hand,using that sα1σw < σw and τ ≤ σw, the lifting property gives sα1τ ≤ sα1σw. Byapplying the induction hypothesis to sα1σ we see that sα1τ = σ′w with σ′ ≤ sα1σ.Hence τ = sα1σ

′w. Finally, σ′ < sα1σ′ (as σ′ is a product of some sαi with i > 1) and

the lifting property shows that sα1σ′ ≤ σ, as required. �

Lemma 6.10. Suppose that w ∈Wρ, σ ∈ Σw.

(i) For any σ′ ≤ σ the representation C(σ′w, σ′w) occurs with multiplicity one inboth PS(w, σw) and (PS(w, σw) )Qp-an.

(ii) The socle of (PS(w, σw) )Qp-an is isomorphic to C(w,w).


Proof. (i) We note that by Proposition 6.6 and its proof the representation PS(w, σw)has a filtration with graded pieces

(Ind

G(Qp)B(Qp) η(σ′′w, σ′′w)

)C0

with σ′′ ≤ σ. Hence

(PS(w, σw) )Qp-an has a filtration with graded pieces PS(σ′′w, σ′′w) with σ′′ ≤ σ. More-over PS(σ′′w, σ′′w) has irreducible constituents C(v, σ′′w) with v ≥ σ′′w. Suppose thatC(σ′w, σ′w) occurs in PS(σ′′w, σ′′w) for some σ′′ ≤ σ. Let τ := σ′w and τ ′ := σ′′w.By Lemma 5.5(ii) we see that τ ≥ τ ′ and W (τ)τ ′ = W (τ)τ .

We claim that τ = τ ′. By Lemma 6.9(ii) we deduce from τ ≥ τ ′ that σ′ ≥ σ′′, soσ′ = uσ′′ with `(σ′) = `(u) + `(σ′′). Hence τ = uτ ′ and by Lemma 6.9(i) we deducethat sατ < τ for any simple reflection sα in the support of u. On the other hand, byabove, u = τ(τ ′)−1 ∈ W (τ), so sατ > τ for any simple reflection sα in the support ofu. This shows that u = 1, so indeed τ = τ ′.

As C(τ, τ) occurs with multiplicity one in PS(τ, τ) we have established part (i) forthe representation (PS(w, σw) )Qp-an.

For the representation PS(w, σw), we first claim that C(τ, τ) ∼= C(τ, σw). Indeed,by Lemma 5.5(ii) this is equivalent to showing that σ(σ′)−1 ∈ W (τ). This is trueby Lemma 6.9(i), which shows that `(σ(σ′)−1) + `(τ) = `(σ(σ′)−1τ) = `(σw). Itfollows that C(τ, τ) occurs in PS(w, σw). As PS(w, σw) injects into (PS(w, σw) )Qp-an

by Proposition 6.6 the proof is complete. (Alternatively we could check directly thatPw,τ (1) = 1 (Kazhdan–Lusztig polynomial) using Jantzen’s criterion [Hum08, §8.7].)

(ii) By the filtration mentioned in (i), if C is any irreducible closed subrepresentationof (PS(w, σw) )Qp-an, then it has to inject into PS(σ′w, σ′w) for some σ′ ≤ σ, henceC ∼= C(σ′w, σ′w) for some σ′ ≤ σ. Using part (i) and that PS(w, σw) injects into(PS(w, σw) )Qp-an we deduce that C injects into PS(w, σw), hence C ∼= C(w, σw). ByLemma 5.5(ii) we have C(w, σw) ∼= C(w,w). Finally, the socle of (PS(w, σw) )Qp-an isirreducible, as C(w,w) occurs with multiplicity one by part (i). �

We need to understand better the ordinary representations Π(ρ)Cρ,w−1 constructed

in [BH15, §3]. To do this, we introduce an abstract framework. Suppose that (I,≤)is a finite poset, that Xi (i ∈ I) are objects of some abelian category A, and that wehave a compatible system of injections Xi ↪→ Xj for any i ≤ j. We say that a subsetJ ⊆ I is a lower subset if i1 ≤ i2 in I and i2 ∈ J imply i1 ∈ J . Consider the followingcondition.

Condition 6.11. For any non-empty lower subset J having upper bound b ∈ I andfor any maximal element m of J we have (

∑J−{m}Xj) ∩Xm =

∑j<mXj inside Xb.

For any lower subset J we define LJ := lim−→j∈J Xj . If J1 ⊆ J2 are two lower subsets,

then we have a canonical map LJ1 → LJ2 . If J = {i ∈ I : i ≤ n} for some n ∈ I,then we write L≤n for LJ and L<n for LJ−{n}. Note that L≤n ∼= Xn. Also note thatL∅ = 0.

Lemma 6.12. The map LJ ′ → LJ is injective for all pairs of lower subsets J ′ ⊆ Jif and only if Condition 6.11 holds. If this holds, then for any lower subsets I1, I2 wehave LI1∩I2 = LI1 ∩ LI2 inside LI .

Proof. We first observe that if I1, I2 are lower subsets, then LI1∪I2∼= LI1 ⊕LI1∩I2 LI2

(write down inverse isomorphisms).


To prove “⇐”, we induct on #J , the case J = ∅ being trivial. We may assume thatJ ′ 6= J . Pick a maximal element m of J which is moreover such that J ′ ⊆ J − {m}(note that m exists since J ′ is a lower subset and that J −{m} is still a lower subset).

Assume first that m is not a maximum of J , then the maps LJ ′ → LJ−{m}, L<m →LJ−{m}, and L<m → L≤m are injective by induction hypothesis (which can be appliedto the latter since L≤m ( J). Hence so is LJ ′ → LJ−{m} ⊕L<m L≤m = LJ and we aredone.

Assume now that m is the maximum of J , so LJ = Xm. We now fix J and inducton #J ′, the case J ′ = ∅ being trivial. Let n denote a maximal element of J ′. If nis a maximum, then Xn = LJ ′ → LJ = Xm is injective by assumption. Otherwise,LJ ′ = LJ ′−{n} ⊕L<n L≤n. By induction the maps

LJ ′−{n} // LJ

L<n //

OO

L≤n

OO

are all injective as previously. The images of LJ ′−{n}, L<n, L≤n inside LJ = Xm areequal to

∑J ′−{n}Xj ,

∑j<nXj , Xn, respectively. Thus the map LJ ′ = LJ ′−{n} ⊕L<n

L≤n → LJ is injective if and only if (∑

J ′−{n}Xj) ∩Xn =∑

j<nXj inside Xm, which

holds by Condition 6.11.As a consequence we know that whenever I1, I2 are lower subsets, the map LI1∪I2 =

LI1 ⊕LI1∩I2 LI2 → LI is injective, so LI1∩I2 = LI1 ∩ LI2 inside LI .To prove “⇒”, we just apply the identity LI1∩I2 = LI1 ∩LI2 inside LI1∪I2 ⊆ LI with

I1 = J − {m} and I2 = {i ∈ I : i ≤ m} (noting that I1 ∪ I2 ⊆ {i ∈ I : i ≤ b}). �

Example 6.13. Suppose that Ci (i ∈ I) are simple objects in A that are pairwise non-isomorphic and that Xi is a finite length object that is multiplicity-free with Jordan–Holder factors {Cj : j ≤ i} such that the submodule structure (i.e. the Alperin diagram)of Xi is described by the partial order ≤. Then Condition 6.11 holds.

The following lemma is in fact already tacitly used in [BH15] (and should have beenproved there!).

Corollary 6.14. Fix w ∈ Wρ, σ ∈ Σw and let Jσ := {α ∈ ∆ : sα ∈ supp(σ)}. Thenthe map Π(ρ)w−1(Jσ) → Π(ρ)Cρ,w−1 is injective (with the notation of [BH15, §3.3], see

(6.8)).

Proof. We apply the above formalism with A the abelian category of admissible con-tinuous representations of G on Banach spaces, I = Σw with respect to ≤, andXσ = Π(ρ)w−1(Jσ) for σ ∈ Σw. Recall that we picked a compatible system of in-jections between the Xσ. To verify Condition 6.11 we fix σ ∈ Σw, playing the role of

the upper bound b ∈ I. For each σ′ ≤ σ we can write Xσ′∼=(

IndG(Qp)PJσ (Qp) Yσ′

)C0

, where

Yσ′ :=(

IndMJσ (Qp)

(MJσ∩PJσ′ )(Qp) Π(ρ)w−1(Jσ′ )

)C0

. The functor F : Y 7→(

IndG(Qp)PJσ (Qp) Y

)C0

from

submodules of Π(ρ)w−1(Jσ) (see Step 2 of the proof of [BH15, Prop. 3.3.3] for the no-tation) to submodules of Π(ρ)w−1(Jσ) respects addition and intersections (for example,

by choosing a continuous section and rewriting F(Y ) ∼= C0(PJσ(Qp)\G(Qp), Y )). Then


Condition 6.11 follows from the corresponding condition on the Levi subgroup MJσ(Qp)by Example 6.13 and [BH15, Rk. 3.3.4(ii)]. �

For any n ∈ I we let Qn := L≤n/L<n.

Lemma 6.15. Suppose that Condition 6.11 holds. Then for any lower subset J , theobject LJ has a filtration with graded pieces isomorphic to Qj (j ∈ J).

Proof. By induction it suffices to show that if J is a lower subset andm ∈ J is a maximalelement, then LJ/LJ−{m} ∼= Qm. To see this, note that the natural map L≤m/L<m →LJ/LJ−{m} is surjective by construction and injective by Lemma 6.12. �

Corollary 6.16. For any w ∈ Wρ the representation Π(ρ)Cρ,w−1 has a filtration with

graded pieces(

IndG(Qp)B(Qp) η(σw, σw)

)C0

with σ ∈ Σw. Moreover we have

socG(Qp)(Π(ρ)Cρ,w−1)Qp-an∼= C(w,w).

Proof. As in the proof of Corollary 6.14 we put ourselves in the context of the aboveformalism. Then the first claim follows from Lemma 6.15. Hence (Π(ρ)Cρ,w−1)Qp-an

has a filtration with graded pieces PS(σw, σw) with σ ∈ Σw. We deduce that if Cis any irreducible closed subrepresentation of (Π(ρ)Cρ,w−1)Qp-an, then it has to inject

into PS(σw, σw) for some σ ∈ Σw, hence C ∼= C(σw, σw) for some σ ∈ Σw. We claimthat C(σw, σw) occurs in (Π(ρ)Cρ,w−1)Qp-an with multiplicity one, or equivalently that

it occurs in PS(σ′w, σ′w) for σ′ ∈ Σw only when σ′ = σ. If C(σw, σw) occurs inPS(σ′w, σ′w), then by Lemma 5.5(ii) we deduce that σw ≥ σ′w, hence σ ≥ σ′ byLemma 6.9(ii). It now follows from the proof of Lemma 6.10(i) that σ = σ′, provingthe claim. Therefore C has to be contained in the subrepresentation (PS(w, σw) )Qp-an

of (Π(ρ)Cρ,w−1)Qp-an, and the claim follows from Lemma 6.10(ii). �

Remark 6.17. Suppose that all representations πσ :=(

IndG(Qp)B(Qp) η(σw, σw)

)C0

with

σ ∈ Σw are (topologically) irreducible. Then the above results show that Π(ρ)Cρ,w−1 is

a multiplicity-free representation with Jordan–Holder factors πσ (σ ∈ Σw) such that thesubmodule structure is described by the poset (Σw,≤). In particular, this establishesthe existence part of Conjecture 3.5.1 in [BH15] (in case all πσ are irreducible). Hauseux[Hau] recently established the uniqueness part (under the same assumption).

Proposition 6.18. With the above assumptions we have that (Π(ρ)ord)Qp-an is isomor-

phic to a subrepresentation of Π(ρ)fs.

Proof. Fix any w ∈Wρ and let C := C(w,w). It suffices to show that the representation

(Π(ρ)Cρ,w−1)Qp-an injects into Π(ρ)fsC . By Corollary 6.16 we know that (Π(ρ)Cρ,w−1)Qp-an

has socle C. By Proposition 6.6 and Corollary 6.14, for each σ ∈ Σw we have aninjection PS(w, σw) ↪→ (Π(ρ)Cρ,w−1)Qp-an, which is unique up to scalars. We also recall

that (w, σw) ∈Wsoc(ρ) for σ ∈ Σw by Lemma 6.3.Step 1: We show that (Π(ρ)Cρ,w−1)Qp-an =

∑σ∈Σw

PS(w, σw). We first consider

n = 2 (with arbitrary ρ satisfying our assumptions) and note that (PS(1, sα))Qp-an =PS(1, sα) + PS(1, 1): by Corollary 6.16 the left-hand side has irreducible constituentsC(1, 1), C(sα, 1), C(sα, sα), each occurring with multiplicity one, and these all occurin the right-hand side.


For general n, by (6.8) it suffices to show (PS(w, σw) )Qp-an =∑

σ′≤σ PS(w, σ′w)for any fixed σ ∈ Σw. We define J , PJ , π as in Proposition 6.6 and its proof, so

that PS(w, σw) ∼= ( IndG(Qp)PJ (Qp) π

)C0

, and by Lemma 2.13 we have (PS(w, σw))Qp-an∼=(

IndG(Qp)PJ (Qp) πQp-an

)Qp-an. By the proof of Proposition 6.6 and by Lemma 2.14, we deduce

that πQp-an is an external completed tensor product of all ψw−1(j)ε−(n−j) for σ(j) = j

and (πi)Qp-an for 1 ≤ i ≤ r, where πi is the representation (6.7) and πi its universalunitary completion (note that πi is admissible by Proposition 6.5). From the previ-ous paragraph we deduce that (πi)Qp-an is the sum of the subrepresentations π and(

IndGL2(Qp)B2(Qp) ψw−1(ni)ε

−(n−ni)⊗ψw−1(ni+1)ε−(n−ni−1)

)Qp-an. The exactness of locally an-

alytic parabolic induction then implies the claim.Step 2: We show that (Π(ρ)Cρ,w−1)Qp-an injects into Π(ρ)fs

C . By Step 1 it follows

that the amalgam A :=⊕

C{PS(w, σw) : σ ∈ Σw} over the common socle C surjectsonto (Π(ρ)Cρ,w−1)Qp-an. As C occurs precisely once in A, namely in the socle, we see

that (Π(ρ)Cρ,w−1)Qp-an is the unique quotient of A that has socle C. We also have that

M(w, σw) := M(w, (σw)Fρ) = PS(w, σw) by Lemma 6.3 and Remark 5.8(ii). Now,

consider the composition A =⊕

C{M(w, σw) : σ ∈ Σw} ↪→ M(ρ)C � Π(ρ)fsC . As

Π(ρ)fsC has socle C, we deduce by what we showed at the beginning of Step 2 that it

factors through a map (Π(ρ)Cρ,w−1)Qp-an → Π(ρ)fsC . By considering socles we see that

it is injective. �

Remark 6.19. The proof shows, in particular, that (Π(ρ)Cρ,w−1)Qp-an for w ∈Wρ can

be described more explicitly as the unique quotient of⊕

C(w,w){PS(w, σw) : σ ∈ Σw}that has socle C(w,w).

Proposition 6.20. For any w ∈Wρ the unitary representation Π(ρ)Cρ,w−1 is the uni-

versal unitary completion of (Π(ρ)Cρ,w−1)Qp-an. Also, Π(ρ)ord is the universal unitary

completion of (Π(ρ)ord)Qp-an.

Proof. Step 1: We show that for any w ∈ Wρ and σ ∈ Σw the unitary representationPS(w, σw) is the universal unitary completion of (PS(w, σw) )Qp-an.

We need to show that if Π is a unitary continuous representation of G(Qp) on a Ba-nach space, then any continuous G(Qp)-equivariant map θ : (PS(w, σw) )Qp-an → Π ex-tends uniquely to a continuous G(Qp)-equivariant map PS(w, σw)→ Π. The unique-ness is clear by the density of locally analytic vectors. Let i : PS(w, σw)→ PS(w, σw) ,i′ : PS(w, σw) → (PS(w, σw) )Qp-an and j : (PS(w, σw) )Qp-an → PS(w, σw) denotethe canonical maps (all of which are injective), so i = j ◦ i′. From the definitionof PS(w, σw) the map θ ◦ i′ extends uniquely to a map θ′ : PS(w, σw) → Π, i.e.θ′ ◦ i = θ◦ i′. It follows that (θ′ ◦j−θ)◦ i′ = 0, i.e. θ′ ◦j−θ factors through the cokernelof i′.

We claim that no irreducible constituent of the cokernel of i′ admits a G(Qp)-invariant O-lattice. This claim easily implies that θ′ ◦ j = θ, completing the proof.Suppose now that C is any irreducible constituent of coker(i′) that admits a G(Qp)-invariant O-lattice. By the proof of Lemma 6.10(i) we know that C ∼= C(τ, σ′w) forsome σ′ ≤ σ and τ ≥ σ′w. More generally, suppose that C(τ, w′) admits a G(Qp)-invariant O-lattice for any (τ, w′) ∈ W 2 with τ ≥ w′. Then the necessary condition ofEmerton (cf. the proof of [Bre16, Cor. 7.7]) shows that (−τ ·(−λ))(t)πB,w′(t) ∈ O for all


t ∈ T+, where πB,w′ := πB,w′Fρ , T+ := {diag(t1, . . . , tn) ∈ T (Qp) : |tit−1

i+1|Qp ≤ 1 ∀i}.(We note that the dot action in [Bre16] is defined relative to B.) By equation [Bre16,

(8.8)] and the line following [Bre16, Rq. 8.7] we deduce that p∑ji=1(hw′−1(i)−hτ−1(i)) ∈ O

for all 1 ≤ j ≤ n, which is easily seen to be equivalent to τ(−h) ≥ w′(−h) by thedominance order on X(T ) relative to B. But −h is dominant with respect to B andτ ≥ w′ by assumption, so τ(−h) ≤ w′(−h). It follows that τ(−h) = w′(−h). Asthe hi are distinct, we deduce that τ = w′. For our constituent C above this meansC ∼= C(σ′w, σ′w). But C does not occur in coker(i′) by Lemma 6.10(i), contradiction.

Step 2: We deduce the result. It is completely formal to see that universal uni-tary completions commute with finite colimits on the additive category of continuousrepresentations of a p-adic reductive group on locally convex vector spaces. (In fact,finite colimits exist in this category, since finite direct sums and cokernels exist.) Simi-larly, the functor of passing to locally Qp-analytic vectors commutes with finite colimitson the abelian category of admissible continuous representations of G(Qp) by [ST03,Thm. 7.1]. Hence from (6.8) we get (Π(ρ)Cρ,w−1)Qp-an

∼= lim−→σ∈Σw(PS(w, σw) )Qp-an and

by Step 1 we deduce((Π(ρ)Cρ,w−1)Qp-an

) ∼= lim−→σ∈Σw

PS(w, σw) ∼= Π(ρ)Cρ,w−1 ,

completing the proof of the first statement. By passing to a finite direct sum over Wρ

we deduce the second statement. �

6.2. Global applications. We give strong evidence to [BH15, Conj. 4.2.2] in the crys-tabelline case (Theorem 6.25).

We keep the global setup and notation of §5.2, but now assume in addition thatp splits completely in F (or equivalently F+). We assume in the following that foreach place v|p of F+ the representation rv = r|Gal(Fv/Fv) is as in §6, namely that it is

upper-triangular as in (6.1), satisfies Hypothesis 5.1, and we choose a representative(still denoted by) rv of rv that is a good conjugate.

Proposition 6.21. Assume Conjecture 5.10. If (walgv , wv)v ∈ W such that walg

v =

walgv (wv) for all v, then restriction to the socle induces an isomorphism

HomG(F+p )

(⊗v|p

PS(walgv , wv)(ε

n−1), S(Up, E)Qp-an[mr])

∼−→ HomG(F+p )

(⊗v|pC(walg

v , wv)(εn−1), S(Up, E)Qp-an[mr]

).

Moreover, any non-zero element of the left-hand side is injective.

Proof. By Remark 5.8(ii) this is a special case of Step 1 of the proof of Theorem 5.12.�


Proposition 6.22. Assume Conjecture 5.10. For each place v|p of F+ suppose thatwv ∈Wrv and σv, σ

′v ∈ Σwv with σv ≤ σ′v. Then the restriction map

(6.23) HomG(F+p )

(⊗v|p

PS(wv, σvwv) (εn−1), S(Up, E)[mr])


(⊗v|p

PS(wv, σ′vwv) (εn−1), S(Up, E)[mr]

)induced by the injections iwv ,σv ,σ′v of §6.1 (see just above (6.8)) is an isomorphism of

finite-dimensional vector spaces. Moreover, any non-zero element of

HomG(F+p )(⊗

v|pPS(wv, σvwv) (εn−1), S(Up, E)[mr]) is injective.

Proof. We will first check the last assertion, by passing to locally Qp-analytic vec-tors, using Lemma 2.14 and [ST03, Thm. 7.1]. Suppose that for each v|p we aregiven an irreducible constituent Cv of (PS(wv, σvwv) )Qp-an. Recall from the proof ofLemma 6.10 that the representation (PS(wv, σvwv) )Qp-an has a filtration with gradedpieces PS(σ′vwv, σ

′vwv) with σ′v ≤ σv, hence by Lemma 5.5(iii) we have Cv ∼= C(τv, σ

′vwv)

for some τv ≥ σ′vwv. If ⊗v|pCv(ε

n−1) injects into S(Up, E)Qp-an[mr], then by Conjec-

ture 5.10, Lemma 6.3 and Lemma 6.9(i) we get that τv ≤ walgv (σ′vwv) = wv ≤ σ′vwv and

hence τv = wv = σ′vwv and σ′v = 1, i.e. Cv ∼= C(wv, wv). From Lemma 2.21 and Lemma

6.10(ii) we deduce that ⊗v|pCv is the G(F+

p )-socle of ⊗v|p

(PS(wv, σvwv) )Qp-an. Therefore,

any non-zero element of HomG(F+p )(⊗

v|pPS(wv, σvwv) (εn−1), S(Up, E)[mr]) is injective.

Since ⊗v|pC(wv, wv) is the G(F+

p )-socle of ⊗v|p

(PS(wv, τvwv) )Qp-an for τv ∈ {σ′v, σv},

occurring as constituent with multiplicity one (by Lemma 6.10(i)), it follows that themap (6.23) is injective.

To complete the proof it suffices to show that the two sides of (6.23) have the samefinite dimension. We note that

(6.24) HomG(F+p )

(⊗v|p


∼= HomG(F+p )

(⊗v|p

PS(wv, σvwv)(εn−1), S(Up, E)Qp-an[mr]

)∼= HomG(F+

p )

(⊗v|pC(wv, σvwv)(ε

n−1), S(Up, E)Qp-an[mr])

by Lemma 3.4, Proposition 6.21, and Lemma 6.3. As C(wv, σvwv) ∼= C(wv, wv) isindependent of σv by Lemma 5.5(ii) and the vector space (6.24) is finite-dimensionalby Lemma 5.15 we complete the proof. �

The theorem that follows gives evidence for [BH15, Conj. 4.2.2] (corrected as in §7below) in the crystabelline case.

Theorem 6.25. Assume Conjecture 5.10. Then there exists an injection of admissible

continuous representations ⊗v|p

Π(rv)ord(εn−1) ↪→ S(Up, E)[mr]. More precisely, for any


w = (wv)v ∈∏v|pWrv let

nw := dimE HomG(F+p )

(⊗v|pC(wv, wv)(ε


Then we have an injection of admissible continuous representations

(6.26)⊕

w=(wv)v

(⊗v|p

Π(rv)Crv ,w−1v

(εn−1))⊕nw ↪→ S(Up, E)[mr].

Proof. By assumption we have an injection

(6.27)⊕

w=(wv)v

(⊗v|pC(wv, wv)(ε

n−1))⊕nw → SQp-an[mr].

Fix any w = (wv)v ∈∏v|pWrv . By the isomorphism (6.8) and Lemma 2.3 we have

HomG(F+p )

(⊗v|p

Π(rv)Crv ,w−1v

(εn−1), S(Up, E)[mr])


(lim−→

σv∈Σwv

⊗v|p


∼= lim←−σv∈Σwv

HomG(F+p )

(⊗v|p

PS(wv, σvwv) (εn−1), S(Up, E)[mr]).

By Proposition 6.22 the projective limit is isomorphic to the final term where σv = 1for all v|p and hence by (6.24) it is further isomorphic to

HomG(F+p )

(⊗v|pC(wv, wv)(ε

n−1), S(Up, E)Qp-an[mr]).

Thus we can extend the map (6.27) uniquely to a map as in (6.26). The extendedmap is injective by the last statement of Corollary 6.16 (using Lemma 2.21) and theinjectivity of (6.27). �

Remark 6.28. Alternatively we could prove Theorem 6.25 using Theorem 5.12, Propo-sition 6.18, and Proposition 6.20. We also recall that Conjecture 5.10 is known in manycases (see Theorem 5.11).

Remark 6.29. When n = 3, some cases of this theorem were claimed in an unpublishedpreprint [BC14].

7. Errata for [BH15]

The definition of Πord just above [BH15, Conj. 4.2.2] should be replaced by thefollowing definition: Πord is the closure (in the admissible continuous representationΠ) of the sum of all its finite length closed subrepresentations with all irreducibleconstituents being constituents of unitary continuous principal series. Then [BH15,

Conj. 4.2.2] can be stated verbatim, and implies in particular that S(Up, E)[pΣ]ord

should be of finite length. Note that, due to the closure process, it is not clear a

priori that all irreducible constituents of S(Up, E)[pΣ]ord are still constituents of unitarycontinuous principal series.

The proof of [BH15, Thm. 4.4.8] is too sketchy and moreover the representation

S(Up, E)pΣ at the end of the proof is not a Banach space (it is not necessarily complete,as it is just some localisation), hence one cannot apply [BH15, Cor. 4.3.11] to it. One


can fix our proof of [BH15, Thm. 4.4.8] (by working instead with the localisation at amaximal ideal of the complete integral Hecke algebra as in [Eme11, §5.2]), but in anycase this result is now a special case of Theorem 6.25 (together with Theorem 5.11).

Finally, due to the above comment on the (corrected) definition of S(Up, E)[pΣ]ord,[BH15, Rk. 4.4.9(a)] should be ignored.

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L.M.O., C.N.R.S., Universite Paris-Sud, Universite Paris-Saclay, 91405 Orsay, FranceE-mail address: [email protected]

Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, CanadaE-mail address: [email protected]

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